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Номер 2.4, страница 22 - гдз по алгебре 11 класс учебник Абылкасымова, Жумагулова
Авторы: Абылкасымова А. Е., Жумагулова З. А.
Тип: Учебник
Издательство: Мектеп
Год издания: 2019 - 2026
Популярные ГДЗ в 11 классе
Глава I. Первообразная и интеграл. Параграф 2. Криволинейная трапеция и её площадь - номер 2.4, страница 22.
App\Models\Task {#1030 // resources/views/models/task/default.blade.php #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1124856 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "22" "field_page_end" => "23" "field_url" => "/11-klass/algebra/mektep-abylkasymova1-uchebnik-2019/02-4" "field_display_title" => "2.4" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ 0 => App\Models\Term {#1036 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1046 #items: array:1 [ 0 => App\Models\Branch {#1045 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1124764 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Первообразная и интеграл" "field_branch_order" => "I" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1056 #items: array:1 [ 0 => App\Models\Term {#1055 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ "field_accusative_case" => "Главу" "field_creative_case" => "Главой" "field_dative_case" => "Главе" "field_genitive_case" => "Главы" "field_nominative_case" => "Глава" "field_prepositional_case" => "Главе" ] "field_page_check_not_needed" => null "field_short_name" => null ] #original: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ "field_accusative_case" => "Главу" "field_creative_case" => "Главой" "field_dative_case" => "Главе" "field_genitive_case" => "Главы" "field_nominative_case" => "Глава" "field_prepositional_case" => "Главе" ] "field_page_check_not_needed" => null "field_short_name" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "10" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1059 #items: array:1 [ 0 => App\Models\Book {#1058 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 4297 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1063 #items: array:1 [ 0 => App\Models\Term {#1060 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ "field_accusative_case" => "алгебру" "field_creative_case" => "алгеброй" "field_dative_case" => "алгебре" "field_genitive_case" => "алгебры" "field_nominative_case" => "алгебра" "field_prepositional_case" => "алгебре" ] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #original: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ "field_accusative_case" => "алгебру" "field_creative_case" => "алгеброй" "field_dative_case" => "алгебре" "field_genitive_case" => "алгебры" "field_nominative_case" => "алгебра" "field_prepositional_case" => "алгебре" ] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1065 #items: array:1 [ 0 => App\Models\Term {#1062 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5460 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "11" "field_cases" => array:6 [ "field_accusative_case" => "одиннадцатый" "field_creative_case" => "одиннадцатым" "field_dative_case" => "одиннадцатому" "field_genitive_case" => "одинадцатого" "field_nominative_case" => "одинадцатый" "field_prepositional_case" => "одиннадцатом" ] "field_translit" => "odinadcatyj" ] #original: array:6 [ "id" => 5460 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "11" "field_cases" => array:6 [ "field_accusative_case" => "одиннадцатый" "field_creative_case" => "одиннадцатым" "field_dative_case" => "одиннадцатому" "field_genitive_case" => "одинадцатого" "field_nominative_case" => "одинадцатый" "field_prepositional_case" => "одиннадцатом" ] "field_translit" => "odinadcatyj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1073 #items: array:1 [ 0 => App\Models\Term {#1072 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6996 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Мектеп" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "mektep" ] #original: array:6 [ "id" => 6996 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Мектеп" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "mektep" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1066 #items: array:2 [ 0 => App\Models\Term {#1077 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 6997 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Абылкасымова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алма" "field_patronymic" => "Есимбековна" "field_surname_rp" => null ] #original: array:12 [ "id" => 6997 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Абылкасымова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алма" "field_patronymic" => "Есимбековна" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Term {#1078 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 6999 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумагулова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Зауре" "field_patronymic" => "Абдыкеновна" "field_surname_rp" => null ] #original: array:12 [ "id" => 6999 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумагулова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Зауре" "field_patronymic" => "Абдыкеновна" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1074 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1070 #items: array:1 [ 0 => App\Models\Term {#1064 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #original: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1068 #items: array:1 [ 0 => App\Models\Term {#1076 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "kazakhstan" ] #original: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "kazakhstan" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1079 #items: array:1 [ 0 => App\Models\Term {#1069 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "almaty" ] #original: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "almaty" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1067 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1061 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1080 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1081 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1082 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1083 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Общественно-гуманитарного направления" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "530" "field_priority" => "2" "field_default_folder" => "/algebra_11/abylkasymova1-u19/" "field_isbn" => null "field_cover" => array:1 [ 0 => "/media/algebra_11/abylkasymova1-u19/covers/cover1.webp?ts=1752223853" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_11/abylkasymova1-u19/covers/cover1.webp?ts=1752223853" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1084 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1085 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1086 #items: [] #escapeWhenCastingToString: false } "field_url" => "/11-klass/algebra/mektep-abylkasymova1-uchebnik-2019" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1087 #items: [] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 382 "subject" => 542 "class_subject" => 388 ] ] #original: array:50 [ "id" => 4297 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1063} "field_class" => Illuminate\Database\Eloquent\Collection {#1065} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1073} "field_author" => Illuminate\Database\Eloquent\Collection {#1066} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1074} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1070} "field_country" => Illuminate\Database\Eloquent\Collection {#1068} "field_city" => Illuminate\Database\Eloquent\Collection {#1079} "field_series" => Illuminate\Database\Eloquent\Collection {#1067} "field_umk" => Illuminate\Database\Eloquent\Collection {#1061} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1080} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1081} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1082} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1083} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Общественно-гуманитарного направления" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "530" "field_priority" => "2" "field_default_folder" => "/algebra_11/abylkasymova1-u19/" "field_isbn" => null "field_cover" => array:1 [ 0 => "/media/algebra_11/abylkasymova1-u19/covers/cover1.webp?ts=1752223853" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_11/abylkasymova1-u19/covers/cover1.webp?ts=1752223853" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1084} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1085} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1086} "field_url" => "/11-klass/algebra/mektep-abylkasymova1-uchebnik-2019" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1087} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 382 "subject" => 542 "class_subject" => 388 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1049 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1124764 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Первообразная и интеграл" "field_branch_order" => "I" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1056} "field_page_start" => "10" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1059} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1049} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1109 #items: array:2 [ 0 => App\Models\Branch {#1117 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1124764 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Первообразная и интеграл" "field_branch_order" => "I" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1118 #items: array:1 [ 0 => App\Models\Term {#1055} ] #escapeWhenCastingToString: false } "field_page_start" => "10" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1119 #items: array:1 [ 0 => App\Models\Book {#1058} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1120 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1124764 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Первообразная и интеграл" "field_branch_order" => "I" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1118} "field_page_start" => "10" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1119} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1120} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Branch {#1121 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1124766 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Криволинейная трапеция и её площадь" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1131 #items: array:1 [ 0 => App\Models\Term {#1130 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ "field_accusative_case" => "Параграф" "field_creative_case" => "Параграфом" "field_dative_case" => "Параграфу" "field_genitive_case" => "Параграфа" "field_nominative_case" => "Параграф" "field_prepositional_case" => "Параграфе" ] "field_page_check_not_needed" => null "field_short_name" => "§" ] #original: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ "field_accusative_case" => "Параграф" "field_creative_case" => "Параграфом" "field_dative_case" => "Параграфу" "field_genitive_case" => "Параграфа" "field_nominative_case" => "Параграф" "field_prepositional_case" => "Параграфе" ] "field_page_check_not_needed" => null "field_short_name" => "§" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "19" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1123 #items: array:1 [ 0 => App\Models\Book {#1058} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1124 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1124766 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Криволинейная трапеция и её площадь" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1131} "field_page_start" => "19" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1123} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1124} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1038 #items: array:2 [ 0 => App\Models\Element {#1100 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 1308093 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1090 #items: array:1 [ 0 => App\Models\Edition {#1099 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 4970 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1097 #items: array:1 [ 0 => App\Models\Term {#1098 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6996 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Мектеп" "field_cases" => array:6 [ …6] "field_translit" => "mektep" ] #original: array:6 [ "id" => 6996 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Мектеп" "field_cases" => array:6 [ …6] "field_translit" => "mektep" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1096 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1094 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1095 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 4970 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1097} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1096} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1094} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1095} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1124856" "type" => "task" ] "text" => "<p><strong>2.4.</strong> Вычислите площади криволинейных трапеций, изображенных на рисунке 14:</p><p><strong>1</strong>)</p><p><svg width="200" height="200" viewBox="-2 -1 4 5"><defs><pattern id="diagonalHatch" patternUnits="userSpaceOnUse" width="0.2" height="0.2" patternTransform="rotate(45)"><line x1="0" y1="0" x2="0" y2="0.2" stroke="gray" stroke-width="0.02" /></pattern></defs><!-- Axes --><line x1="-2" y1="0" x2="2" y2="0" stroke="black" stroke-width="0.05" /><line x1="0" y1="-1" x2="0" y2="5" stroke="black" stroke-width="0.05" /><!-- Arrows --><path d="M 1.9 0 L 1.8 0.05 M 1.9 0 L 1.8 -0.05" stroke="black" stroke-width="0.05" /><path d="M 0 4.9 L 0.05 4.8 M 0 4.9 L -0.05 4.8" stroke="black" stroke-width="0.05" /><!-- Labels --><text x="1.9" y="-0.2" font-size="0.2">x</text><text x="0.1" y="4.8" font-size="0.2">y</text><text x="-0.05" y="-0.05" font-size="0.2">O</text><!-- Ticks and labels --><line x1="-1" y1="-0.05" x2="-1" y2="0.05" stroke="black" stroke-width="0.05" /><text x="-1" y="-0.2" text-anchor="middle" font-size="0.2">-1</text><line x1="1" y1="-0.05" x2="1" y2="0.05" stroke="black" stroke-width="0.05" /><text x="1" y="-0.2" text-anchor="middle" font-size="0.2">1</text><line x1="-0.05" y1="4" x2="0.05" y2="4" stroke="black" stroke-width="0.05" /><text x="-0.2" y="4" text-anchor="end" font-size="0.2">4</text><!-- Curve: y = (x-1)^2 --><path d="M -1 4 Q 0 1 1 0 L 2 1" fill="none" stroke="black" stroke-width="0.08" /><!-- Curve Label --><text x="1.5" y="1.5" transform="rotate(45 1.5,1.5)" font-size="0.2">$y=x^2-2x+1$</text><!-- Shaded region: From x=-1 to x=1 --><path d="M -1 0 L 1 0 Q 0 1 -1 4 Z" fill="url(#diagonalHatch)" stroke="none" /></svg></p><p>$y = x^2 - 2x + 1$</p><p><strong>2</strong>)</p><p><svg width="200" height="200" viewBox="-1 -0.5 4 4.5"><defs><pattern id="horizontalHatch" patternUnits="userSpaceOnUse" width="0.2" height="0.2" patternTransform="rotate(0)"><line x1="0" y1="0" x2="0.2" y2="0" stroke="gray" stroke-width="0.02" /></pattern></defs><!-- Axes --><line x1="-1" y1="0" x2="3" y2="0" stroke="black" stroke-width="0.05" /><line x1="0" y1="-0.5" x2="0" y2="4" stroke="black" stroke-width="0.05" /><!-- Arrows --><path d="M 2.9 0 L 2.8 0.05 M 2.9 0 L 2.8 -0.05" stroke="black" stroke-width="0.05" /><path d="M 0 3.9 L 0.05 3.8 M 0 3.9 L -0.05 3.8" stroke="black" stroke-width="0.05" /><!-- Labels --><text x="2.9" y="-0.2" font-size="0.2">x</text><text x="0.1" y="3.8" font-size="0.2">y</text><text x="-0.05" y="-0.05" font-size="0.2">O</text><!-- Ticks and labels --><line x1="1" y1="-0.05" x2="1" y2="0.05" stroke="black" stroke-width="0.05" /><text x="1" y="-0.2" text-anchor="middle" font-size="0.2">1</text><line x1="2" y1="-0.05" x2="2" y2="0.05" stroke="black" stroke-width="0.05" /><text x="2" y="-0.2" text-anchor="middle" font-size="0.2">2</text><line x1="-0.05" y1="3" x2="0.05" y2="3" stroke="black" stroke-width="0.05" /><text x="-0.2" y="3" text-anchor="end" font-size="0.2">3</text><!-- Curve: y = 2 + 2x - x^2 --><path d="M -0.5 0.75 L 0 2 L 1 3 L 2 2 L 2.5 0.75" fill="none" stroke="black" stroke-width="0.08" /><!-- Curve Label --><text x="1.5" y="2.5" font-size="0.2">$y=2+2x-x^2$</text><!-- Shaded region: From x=0 to x=2 --><path d="M 0 0 L 2 0 L 2 2 Q 1 3 0 2 Z" fill="url(#horizontalHatch)" stroke="none" /></svg></p><p>$y = 2 + 2x - x^2$</p><p><strong>3</strong>)</p><p><svg width="200" height="200" viewBox="-3 -1 6 6"><defs><pattern id="diagonalHatch" patternUnits="userSpaceOnUse" width="0.2" height="0.2" patternTransform="rotate(45)"><line x1="0" y1="0" x2="0" y2="0.2" stroke="gray" stroke-width="0.02" /></pattern></defs><!-- Axes --><line x1="-3" y1="0" x2="3" y2="0" stroke="black" stroke-width="0.05" /><line x1="0" y1="-1" x2="0" y2="5" stroke="black" stroke-width="0.05" /><!-- Arrows --><path d="M 2.9 0 L 2.8 0.05 M 2.9 0 L 2.8 -0.05" stroke="black" stroke-width="0.05" /><path d="M 0 4.9 L 0.05 4.8 M 0 4.9 L -0.05 4.8" stroke="black" stroke-width="0.05" /><!-- Labels --><text x="2.9" y="-0.2" font-size="0.2">x</text><text x="0.1" y="4.8" font-size="0.2">y</text><text x="-0.05" y="-0.05" font-size="0.2">O</text><!-- Ticks and labels --><line x1="-2" y1="-0.05" x2="-2" y2="0.05" stroke="black" stroke-width="0.05" /><text x="-2" y="-0.2" text-anchor="middle" font-size="0.2">-2</text><line x1="2" y1="-0.05" x2="2" y2="0.05" stroke="black" stroke-width="0.05" /><text x="2" y="-0.2" text-anchor="middle" font-size="0.2">2</text><line x1="-0.05" y1="4" x2="0.05" y2="4" stroke="black" stroke-width="0.05" /><text x="-0.2" y="4" text-anchor="end" font-size="0.2">4</text><!-- Curve: y = x^2 --><path d="M -2 4 Q 0 0 2 4" fill="none" stroke="black" stroke-width="0.08" /><!-- Curve Label --><text x="1.5" y="2.5" font-size="0.2">$y=x^2$</text><!-- Shaded region: From x=-2 to x=2 --><path d="M -2 0 L 2 0 L 2 4 Q 0 0 -2 4 Z" fill="url(#diagonalHatch)" stroke="none" /></svg></p><p>$y = x^2$</p><p><strong>4</strong>)</p><p><svg width="200" height="200" viewBox="-0.5 -0.5 4 4.5"><defs><pattern id="diagonalHatch" patternUnits="userSpaceOnUse" width="0.2" height="0.2" patternTransform="rotate(45)"><line x1="0" y1="0" x2="0" y2="0.2" stroke="gray" stroke-width="0.02" /></pattern></defs><!-- Axes --><line x1="-0.5" y1="0" x2="3.5" y2="0" stroke="black" stroke-width="0.05" /><line x1="0" y1="-0.5" x2="0" y2="4" stroke="black" stroke-width="0.05" /><!-- Arrows --><path d="M 3.4 0 L 3.3 0.05 M 3.4 0 L 3.3 -0.05" stroke="black" stroke-width="0.05" /><path d="M 0 3.9 L 0.05 3.8 M 0 3.9 L -0.05 3.8" stroke="black" stroke-width="0.05" /><!-- Labels --><text x="3.4" y="-0.2" font-size="0.2">x</text><text x="0.1" y="3.8" font-size="0.2">y</text><text x="-0.05" y="-0.05" font-size="0.2">O</text><!-- Ticks and labels --><line x1="0.785" y1="-0.05" x2="0.785" y2="0.05" stroke="black" stroke-width="0.05" /><text x="0.785" y="-0.3" text-anchor="middle" font-size="0.2">$\frac{\pi}{4}$</text><line x1="1.57" y1="-0.05" x2="1.57" y2="0.05" stroke="black" stroke-width="0.05" /><text x="1.57" y="-0.3" text-anchor="middle" font-size="0.2">$\frac{\pi}{2}$</text><line x1="3.14" y1="-0.05" x2="3.14" y2="0.05" stroke="black" stroke-width="0.05" /><text x="3.14" y="-0.2" text-anchor="middle" font-size="0.2">$\pi$</text><line x1="-0.05" y1="3" x2="0.05" y2="3" stroke="black" stroke-width="0.05" /><text x="-0.2" y="3" text-anchor="end" font-size="0.2">3</text><!-- Curve: y = 3sin(x) --><path d="M 0 0 C 0.5 3 2.5 3 3.14 0" fill="none" stroke="black" stroke-width="0.08" /><!-- Curve Label --><text x="2.5" y="2.5" font-size="0.2">$y=3\sin x$</text><!-- Shaded region: From x=0 to x=pi --><path d="M 0 0 L 3.14 0 C 2.5 3 0.5 3 0 0 Z" fill="url(#diagonalHatch)" stroke="none" /></svg></p><p>$y = 3\sin x$</p><p>Рис. 14</p>" "img" => array:2 [ 0 => array:5 [ "name" => "4-1.jpg" "alt" => null "width" => "1785" "height" => 227 "path" => "/media/algebra_11/abylkasymova1-u19/0-02/4-1.webp?ts=1753819172" ] 1 => array:5 [ "name" => "4-2.jpg" "alt" => null "width" => "1631" "height" => 1271 "path" => "/media/algebra_11/abylkasymova1-u19/0-02/4-2.webp?ts=1753819172" ] ] ] #original: array:7 [ "id" => 1308093 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1090} "task" => array:2 [ "refs" => "1124856" "type" => "task" ] "text" => "<p><strong>2.4.</strong> Вычислите площади криволинейных трапеций, изображенных на рисунке 14:</p><p><strong>1</strong>)</p><p><svg width="200" height="200" viewBox="-2 -1 4 5"><defs><pattern id="diagonalHatch" patternUnits="userSpaceOnUse" width="0.2" height="0.2" patternTransform="rotate(45)"><line x1="0" y1="0" x2="0" y2="0.2" stroke="gray" stroke-width="0.02" /></pattern></defs><!-- Axes --><line x1="-2" y1="0" x2="2" y2="0" stroke="black" stroke-width="0.05" /><line x1="0" y1="-1" x2="0" y2="5" stroke="black" stroke-width="0.05" /><!-- Arrows --><path d="M 1.9 0 L 1.8 0.05 M 1.9 0 L 1.8 -0.05" stroke="black" stroke-width="0.05" /><path d="M 0 4.9 L 0.05 4.8 M 0 4.9 L -0.05 4.8" stroke="black" stroke-width="0.05" /><!-- Labels --><text x="1.9" y="-0.2" font-size="0.2">x</text><text x="0.1" y="4.8" font-size="0.2">y</text><text x="-0.05" y="-0.05" font-size="0.2">O</text><!-- Ticks and labels --><line x1="-1" y1="-0.05" x2="-1" y2="0.05" stroke="black" stroke-width="0.05" /><text x="-1" y="-0.2" text-anchor="middle" font-size="0.2">-1</text><line x1="1" y1="-0.05" x2="1" y2="0.05" stroke="black" stroke-width="0.05" /><text x="1" y="-0.2" text-anchor="middle" font-size="0.2">1</text><line x1="-0.05" y1="4" x2="0.05" y2="4" stroke="black" stroke-width="0.05" /><text x="-0.2" y="4" text-anchor="end" font-size="0.2">4</text><!-- Curve: y = (x-1)^2 --><path d="M -1 4 Q 0 1 1 0 L 2 1" fill="none" stroke="black" stroke-width="0.08" /><!-- Curve Label --><text x="1.5" y="1.5" transform="rotate(45 1.5,1.5)" font-size="0.2">$y=x^2-2x+1$</text><!-- Shaded region: From x=-1 to x=1 --><path d="M -1 0 L 1 0 Q 0 1 -1 4 Z" fill="url(#diagonalHatch)" stroke="none" /></svg></p><p>$y = x^2 - 2x + 1$</p><p><strong>2</strong>)</p><p><svg width="200" height="200" viewBox="-1 -0.5 4 4.5"><defs><pattern id="horizontalHatch" patternUnits="userSpaceOnUse" width="0.2" height="0.2" patternTransform="rotate(0)"><line x1="0" y1="0" x2="0.2" y2="0" stroke="gray" stroke-width="0.02" /></pattern></defs><!-- Axes --><line x1="-1" y1="0" x2="3" y2="0" stroke="black" stroke-width="0.05" /><line x1="0" y1="-0.5" x2="0" y2="4" stroke="black" stroke-width="0.05" /><!-- Arrows --><path d="M 2.9 0 L 2.8 0.05 M 2.9 0 L 2.8 -0.05" stroke="black" stroke-width="0.05" /><path d="M 0 3.9 L 0.05 3.8 M 0 3.9 L -0.05 3.8" stroke="black" stroke-width="0.05" /><!-- Labels --><text x="2.9" y="-0.2" font-size="0.2">x</text><text x="0.1" y="3.8" font-size="0.2">y</text><text x="-0.05" y="-0.05" font-size="0.2">O</text><!-- Ticks and labels --><line x1="1" y1="-0.05" x2="1" y2="0.05" stroke="black" stroke-width="0.05" /><text x="1" y="-0.2" text-anchor="middle" font-size="0.2">1</text><line x1="2" y1="-0.05" x2="2" y2="0.05" stroke="black" stroke-width="0.05" /><text x="2" y="-0.2" text-anchor="middle" font-size="0.2">2</text><line x1="-0.05" y1="3" x2="0.05" y2="3" stroke="black" stroke-width="0.05" /><text x="-0.2" y="3" text-anchor="end" font-size="0.2">3</text><!-- Curve: y = 2 + 2x - x^2 --><path d="M -0.5 0.75 L 0 2 L 1 3 L 2 2 L 2.5 0.75" fill="none" stroke="black" stroke-width="0.08" /><!-- Curve Label --><text x="1.5" y="2.5" font-size="0.2">$y=2+2x-x^2$</text><!-- Shaded region: From x=0 to x=2 --><path d="M 0 0 L 2 0 L 2 2 Q 1 3 0 2 Z" fill="url(#horizontalHatch)" stroke="none" /></svg></p><p>$y = 2 + 2x - x^2$</p><p><strong>3</strong>)</p><p><svg width="200" height="200" viewBox="-3 -1 6 6"><defs><pattern id="diagonalHatch" patternUnits="userSpaceOnUse" width="0.2" height="0.2" patternTransform="rotate(45)"><line x1="0" y1="0" x2="0" y2="0.2" stroke="gray" stroke-width="0.02" /></pattern></defs><!-- Axes --><line x1="-3" y1="0" x2="3" y2="0" stroke="black" stroke-width="0.05" /><line x1="0" y1="-1" x2="0" y2="5" stroke="black" stroke-width="0.05" /><!-- Arrows --><path d="M 2.9 0 L 2.8 0.05 M 2.9 0 L 2.8 -0.05" stroke="black" stroke-width="0.05" /><path d="M 0 4.9 L 0.05 4.8 M 0 4.9 L -0.05 4.8" stroke="black" stroke-width="0.05" /><!-- Labels --><text x="2.9" y="-0.2" font-size="0.2">x</text><text x="0.1" y="4.8" font-size="0.2">y</text><text x="-0.05" y="-0.05" font-size="0.2">O</text><!-- Ticks and labels --><line x1="-2" y1="-0.05" x2="-2" y2="0.05" stroke="black" stroke-width="0.05" /><text x="-2" y="-0.2" text-anchor="middle" font-size="0.2">-2</text><line x1="2" y1="-0.05" x2="2" y2="0.05" stroke="black" stroke-width="0.05" /><text x="2" y="-0.2" text-anchor="middle" font-size="0.2">2</text><line x1="-0.05" y1="4" x2="0.05" y2="4" stroke="black" stroke-width="0.05" /><text x="-0.2" y="4" text-anchor="end" font-size="0.2">4</text><!-- Curve: y = x^2 --><path d="M -2 4 Q 0 0 2 4" fill="none" stroke="black" stroke-width="0.08" /><!-- Curve Label --><text x="1.5" y="2.5" font-size="0.2">$y=x^2$</text><!-- Shaded region: From x=-2 to x=2 --><path d="M -2 0 L 2 0 L 2 4 Q 0 0 -2 4 Z" fill="url(#diagonalHatch)" stroke="none" /></svg></p><p>$y = x^2$</p><p><strong>4</strong>)</p><p><svg width="200" height="200" viewBox="-0.5 -0.5 4 4.5"><defs><pattern id="diagonalHatch" patternUnits="userSpaceOnUse" width="0.2" height="0.2" patternTransform="rotate(45)"><line x1="0" y1="0" x2="0" y2="0.2" stroke="gray" stroke-width="0.02" /></pattern></defs><!-- Axes --><line x1="-0.5" y1="0" x2="3.5" y2="0" stroke="black" stroke-width="0.05" /><line x1="0" y1="-0.5" x2="0" y2="4" stroke="black" stroke-width="0.05" /><!-- Arrows --><path d="M 3.4 0 L 3.3 0.05 M 3.4 0 L 3.3 -0.05" stroke="black" stroke-width="0.05" /><path d="M 0 3.9 L 0.05 3.8 M 0 3.9 L -0.05 3.8" stroke="black" stroke-width="0.05" /><!-- Labels --><text x="3.4" y="-0.2" font-size="0.2">x</text><text x="0.1" y="3.8" font-size="0.2">y</text><text x="-0.05" y="-0.05" font-size="0.2">O</text><!-- Ticks and labels --><line x1="0.785" y1="-0.05" x2="0.785" y2="0.05" stroke="black" stroke-width="0.05" /><text x="0.785" y="-0.3" text-anchor="middle" font-size="0.2">$\frac{\pi}{4}$</text><line x1="1.57" y1="-0.05" x2="1.57" y2="0.05" stroke="black" stroke-width="0.05" /><text x="1.57" y="-0.3" text-anchor="middle" font-size="0.2">$\frac{\pi}{2}$</text><line x1="3.14" y1="-0.05" x2="3.14" y2="0.05" stroke="black" stroke-width="0.05" /><text x="3.14" y="-0.2" text-anchor="middle" font-size="0.2">$\pi$</text><line x1="-0.05" y1="3" x2="0.05" y2="3" stroke="black" stroke-width="0.05" /><text x="-0.2" y="3" text-anchor="end" font-size="0.2">3</text><!-- Curve: y = 3sin(x) --><path d="M 0 0 C 0.5 3 2.5 3 3.14 0" fill="none" stroke="black" stroke-width="0.08" /><!-- Curve Label --><text x="2.5" y="2.5" font-size="0.2">$y=3\sin x$</text><!-- Shaded region: From x=0 to x=pi --><path d="M 0 0 L 3.14 0 C 2.5 3 0.5 3 0 0 Z" fill="url(#diagonalHatch)" stroke="none" /></svg></p><p>$y = 3\sin x$</p><p>Рис. 14</p>" "img" => array:2 [ 0 => array:5 [ "name" => "4-1.jpg" "alt" => null "width" => "1785" "height" => 227 "path" => "/media/algebra_11/abylkasymova1-u19/0-02/4-1.webp?ts=1753819172" ] 1 => array:5 [ "name" => "4-2.jpg" "alt" => null "width" => "1631" "height" => 1271 "path" => "/media/algebra_11/abylkasymova1-u19/0-02/4-2.webp?ts=1753819172" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Element {#1092 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1553102 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1047 #items: array:1 [ 0 => App\Models\Edition {#1093 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 5663 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1089 #items: array:1 [ 0 => App\Models\Term {#1091 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 7012 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Gemini 2.5 Pro" "field_cases" => array:6 [ …6] "field_translit" => "gemini 2.5 pro" ] #original: array:6 [ "id" => 7012 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Gemini 2.5 Pro" "field_cases" => array:6 [ …6] "field_translit" => "gemini 2.5 pro" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1051 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1088 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1050 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5663 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1089} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1051} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1088} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1050} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1124856" "type" => "task" ] "text" => "<p><strong>1</strong>)</p><p>Площадь криволинейной трапеции, ограниченной графиком функции $y=f(x)$, осью $Ox$ и прямыми $x=a$ и $x=b$, вычисляется по формуле Ньютона-Лейбница: $S = \int_a^b f(x)dx$.</p><p>В данном случае функция $f(x) = x^2 - 2x + 1$. Из графика видно, что пределы интегрирования: $a=0$ и $b=1$.</p><p>Вычислим определенный интеграл:</p><p>$S = \int_0^1 (x^2 - 2x + 1)dx$</p><p>Найдем первообразную для функции $f(x) = x^2 - 2x + 1$:</p><p>$F(x) = \frac{x^3}{3} - 2\frac{x^2}{2} + x = \frac{x^3}{3} - x^2 + x$</p><p>Теперь применим формулу Ньютона-Лейбница:</p><p>$S = F(1) - F(0) = \left. \left(\frac{x^3}{3} - x^2 + x \right) \right|_0^1 = \left(\frac{1^3}{3} - 1^2 + 1 \right) - \left(\frac{0^3}{3} - 0^2 + 0 \right) = \left(\frac{1}{3} - 1 + 1 \right) - 0 = \frac{1}{3}$</p><p><strong>Ответ:</strong> $\frac{1}{3}$</p><p><strong>2</strong>)</p><p>Функция, ограничивающая трапецию сверху: $y = 2 + 2x - x^2$. Пределы интегрирования по оси $Ox$, согласно рисунку, от $a=0$ до $b=2$.</p><p>Вычислим площадь с помощью определенного интеграла:</p><p>$S = \int_0^2 (2 + 2x - x^2)dx$</p><p>Находим первообразную:</p><p>$F(x) = 2x + 2\frac{x^2}{2} - \frac{x^3}{3} = 2x + x^2 - \frac{x^3}{3}$</p><p>Вычисляем площадь:</p><p>$S = F(2) - F(0) = \left. \left(2x + x^2 - \frac{x^3}{3} \right) \right|_0^2 = \left(2 \cdot 2 + 2^2 - \frac{2^3}{3} \right) - \left(2 \cdot 0 + 0^2 - \frac{0^3}{3} \right) = \left(4 + 4 - \frac{8}{3} \right) - 0 = 8 - \frac{8}{3} = \frac{24 - 8}{3} = \frac{16}{3}$</p><p><strong>Ответ:</strong> $\frac{16}{3}$</p><p><strong>3</strong>)</p><p>Фигура ограничена графиком функции $y = \frac{1}{4}x^4$ и прямыми $x=-2$ и $x=2$.</p><p>Функция $f(x) = \frac{1}{4}x^4$ является четной, так как $f(-x) = \frac{1}{4}(-x)^4 = \frac{1}{4}x^4 = f(x)$. Интервал интегрирования $[-2, 2]$ симметричен относительно нуля. Поэтому площадь можно вычислить как удвоенную площадь на отрезке $[0, 2]$:</p><p>$S = \int_{-2}^2 \frac{1}{4}x^4 dx = 2 \int_0^2 \frac{1}{4}x^4 dx = \frac{1}{2} \int_0^2 x^4 dx$</p><p>Находим первообразную для $x^4$:</p><p>$F(x) = \frac{x^5}{5}$</p><p>Вычисляем площадь:</p><p>$S = \frac{1}{2} \left. \left(\frac{x^5}{5} \right) \right|_0^2 = \frac{1}{2} \left(\frac{2^5}{5} - \frac{0^5}{5} \right) = \frac{1}{2} \left(\frac{32}{5} - 0 \right) = \frac{32}{10} = \frac{16}{5}$</p><p><strong>Ответ:</strong> $\frac{16}{5}$</p><p><strong>4</strong>)</p><p>Фигура ограничена графиком функции $y = 3\sin x$ и прямыми $x=\frac{\pi}{4}$ и $x=\pi$.</p><p>Вычисляем площадь с помощью определенного интеграла:</p><p>$S = \int_{\pi/4}^{\pi} 3\sin x dx$</p><p>Находим первообразную для $3\sin x$:</p><p>$F(x) = -3\cos x$</p><p>Вычисляем площадь:</p><p>$S = \left. (-3\cos x) \right|_{\pi/4}^{\pi} = (-3\cos(\pi)) - (-3\cos(\frac{\pi}{4})) = -3(-1) + 3\left(\frac{\sqrt{2}}{2}\right) = 3 + \frac{3\sqrt{2}}{2}$</p><p><strong>Ответ:</strong> $3 + \frac{3\sqrt{2}}{2}$</p>" ] #original: array:6 [ "id" => 1553102 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1047} "task" => array:2 [ "refs" => "1124856" "type" => "task" ] "text" => "<p><strong>1</strong>)</p><p>Площадь криволинейной трапеции, ограниченной графиком функции $y=f(x)$, осью $Ox$ и прямыми $x=a$ и $x=b$, вычисляется по формуле Ньютона-Лейбница: $S = \int_a^b f(x)dx$.</p><p>В данном случае функция $f(x) = x^2 - 2x + 1$. Из графика видно, что пределы интегрирования: $a=0$ и $b=1$.</p><p>Вычислим определенный интеграл:</p><p>$S = \int_0^1 (x^2 - 2x + 1)dx$</p><p>Найдем первообразную для функции $f(x) = x^2 - 2x + 1$:</p><p>$F(x) = \frac{x^3}{3} - 2\frac{x^2}{2} + x = \frac{x^3}{3} - x^2 + x$</p><p>Теперь применим формулу Ньютона-Лейбница:</p><p>$S = F(1) - F(0) = \left. \left(\frac{x^3}{3} - x^2 + x \right) \right|_0^1 = \left(\frac{1^3}{3} - 1^2 + 1 \right) - \left(\frac{0^3}{3} - 0^2 + 0 \right) = \left(\frac{1}{3} - 1 + 1 \right) - 0 = \frac{1}{3}$</p><p><strong>Ответ:</strong> $\frac{1}{3}$</p><p><strong>2</strong>)</p><p>Функция, ограничивающая трапецию сверху: $y = 2 + 2x - x^2$. Пределы интегрирования по оси $Ox$, согласно рисунку, от $a=0$ до $b=2$.</p><p>Вычислим площадь с помощью определенного интеграла:</p><p>$S = \int_0^2 (2 + 2x - x^2)dx$</p><p>Находим первообразную:</p><p>$F(x) = 2x + 2\frac{x^2}{2} - \frac{x^3}{3} = 2x + x^2 - \frac{x^3}{3}$</p><p>Вычисляем площадь:</p><p>$S = F(2) - F(0) = \left. \left(2x + x^2 - \frac{x^3}{3} \right) \right|_0^2 = \left(2 \cdot 2 + 2^2 - \frac{2^3}{3} \right) - \left(2 \cdot 0 + 0^2 - \frac{0^3}{3} \right) = \left(4 + 4 - \frac{8}{3} \right) - 0 = 8 - \frac{8}{3} = \frac{24 - 8}{3} = \frac{16}{3}$</p><p><strong>Ответ:</strong> $\frac{16}{3}$</p><p><strong>3</strong>)</p><p>Фигура ограничена графиком функции $y = \frac{1}{4}x^4$ и прямыми $x=-2$ и $x=2$.</p><p>Функция $f(x) = \frac{1}{4}x^4$ является четной, так как $f(-x) = \frac{1}{4}(-x)^4 = \frac{1}{4}x^4 = f(x)$. Интервал интегрирования $[-2, 2]$ симметричен относительно нуля. Поэтому площадь можно вычислить как удвоенную площадь на отрезке $[0, 2]$:</p><p>$S = \int_{-2}^2 \frac{1}{4}x^4 dx = 2 \int_0^2 \frac{1}{4}x^4 dx = \frac{1}{2} \int_0^2 x^4 dx$</p><p>Находим первообразную для $x^4$:</p><p>$F(x) = \frac{x^5}{5}$</p><p>Вычисляем площадь:</p><p>$S = \frac{1}{2} \left. \left(\frac{x^5}{5} \right) \right|_0^2 = \frac{1}{2} \left(\frac{2^5}{5} - \frac{0^5}{5} \right) = \frac{1}{2} \left(\frac{32}{5} - 0 \right) = \frac{32}{10} = \frac{16}{5}$</p><p><strong>Ответ:</strong> $\frac{16}{5}$</p><p><strong>4</strong>)</p><p>Фигура ограничена графиком функции $y = 3\sin x$ и прямыми $x=\frac{\pi}{4}$ и $x=\pi$.</p><p>Вычисляем площадь с помощью определенного интеграла:</p><p>$S = \int_{\pi/4}^{\pi} 3\sin x dx$</p><p>Находим первообразную для $3\sin x$:</p><p>$F(x) = -3\cos x$</p><p>Вычисляем площадь:</p><p>$S = \left. (-3\cos x) \right|_{\pi/4}^{\pi} = (-3\cos(\pi)) - (-3\cos(\frac{\pi}{4})) = -3(-1) + 3\left(\frac{\sqrt{2}}{2}\right) = 3 + \frac{3\sqrt{2}}{2}$</p><p><strong>Ответ:</strong> $3 + \frac{3\sqrt{2}}{2}$</p>" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1124857" "type" => "task" ] "previous" => array:2 [ "refs" => "1124855" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1103 #items: array:1 [ 0 => App\Models\Book {#1058} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1040 #items: array:1 [ 0 => App\Models\BookPage {#1104 #connection: "mysql" #table: "book_pages" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 1100000 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "22" "field_url" => "/11-klass/algebra/mektep-abylkasymova1-uchebnik-2019/page-22" "field_display_title" => "22" "field_folder" => "1" "field_image_name" => "22" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1053 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1052 #items: array:1 [ 0 => App\Models\Book {#1058} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "edition_groups" => Illuminate\Database\Eloquent\Collection {#1048 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1113 #items: array:1 [ 0 => App\Models\Element {#1110 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1263424 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1111 #items: array:1 [ 0 => App\Models\Edition {#1099} ] #escapeWhenCastingToString: false } "book_page" => array:2 [ "refs" => "1100000" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "22-1.jpg" "alt" => null "width" => "2009" "height" => 2912 "path" => "/media/algebra_11/abylkasymova1-u19/0-1/22-1.webp?ts=1752224201" ] ] ] #original: array:6 [ "id" => 1263424 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1111} "book_page" => array:2 [ "refs" => "1100000" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "22-1.jpg" "alt" => null "width" => "2009" "height" => 2912 "path" => "/media/algebra_11/abylkasymova1-u19/0-1/22-1.webp?ts=1752224201" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1100001" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1099999" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1186 #items: array:5 [ 0 => App\Models\Task {#1198 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1124852 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "22" "field_page_end" => null "field_url" => "/11-klass/algebra/mektep-abylkasymova1-uchebnik-2019/vopr-2" "field_display_title" => "Вопросы" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1199 #items: [] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1200 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1201 #items: array:1 [ 0 => App\Models\Branch {#1117} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1202 #items: array:2 [ 0 => App\Models\Branch {#1117} 1 => App\Models\Branch {#1121} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1204 #items: array:2 [ 0 => App\Models\Element {#1213 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 1308572 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1214 …2} "task" => array:2 [ …2] "text" => "<svg width="24" height="24" viewBox="0 0 24 24" fill="none" xmlns="http://www.w3.org/2000/svg"><circle cx="12" cy="12" r="10" stroke="#008080" stroke-width="2"/><path d="M12 15.98V17.98H10V15.98H12ZM10 6.98H12C13.2 6.98 14.2 7.88 14.2 9.08C14.2 9.58 14 10.08 13.7 10.38C13.4 10.68 12.9 10.98 12.4 11.18L11.7 11.48V13.98H10V12.38C10 12.08 10.1 11.78 10.3 11.48C10.5 11.18 10.8 10.88 11.1 10.58L11.8 10.28C12.3 10.08 12.6 9.88 12.8 9.68C13 9.48 13.1 9.28 13.1 8.98C13.1 8.68 13 8.38 12.8 8.18C12.6 7.98 12.3 7.88 12 7.88H10V6.98Z" fill="#008080"/></svg><p><strong>1.</strong> Чем отличается понятие криволинейной трапеции от понятия трапеции?</p><p><strong>2.</strong> Можно ли вычислить площадь криволинейной трапеции с помощью известных формул из геометрии?</p><p><strong>3.</strong> Какой должна быть функция $y = f(x)$ на отрезке $[a; b]$?</p>" "img" => array:1 [ …1] ] #original: array:7 [ "id" => 1308572 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1214 …2} "task" => array:2 [ …2] …2 ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 1 => App\Models\Element {#1215 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1124853" "type" => "task" ] "previous" => array:2 [ "refs" => "1124851" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1210 #items: array:1 [ 0 => App\Models\Book {#1058} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1100000" "type" => "book_page" ] ] #original: array:24 [ "id" => 1124852 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "22" "field_page_end" => null "field_url" => "/11-klass/algebra/mektep-abylkasymova1-uchebnik-2019/vopr-2" "field_display_title" => "Вопросы" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1199} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1200} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1201} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1202} "content" => Illuminate\Database\Eloquent\Collection {#1204} "next" => array:2 [ "refs" => "1124853" "type" => "task" ] "previous" => array:2 [ "refs" => "1124851" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1210} "page" => array:2 [ "refs" => "1100000" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Task {#1205 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1124853 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "22" "field_page_end" => null "field_url" => "/11-klass/algebra/mektep-abylkasymova1-uchebnik-2019/02-1" "field_display_title" => "2.1" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1203 #items: array:1 [ 0 => App\Models\Term {#1036} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1212 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1208 #items: array:1 [ 0 => App\Models\Branch {#1117} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1206 #items: array:2 [ 0 => App\Models\Branch {#1117} 1 => App\Models\Branch {#1121} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1207 #items: array:2 [ 0 => App\Models\Element {#1225 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ …7] #original: array:7 [ …7] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 1 => App\Models\Element {#1227 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1124854" "type" => "task" ] "previous" => array:2 [ "refs" => "1124852" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1222 #items: array:1 [ 0 => App\Models\Book {#1058} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1100000" "type" => "book_page" ] ] #original: array:24 [ "id" => 1124853 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "22" "field_page_end" => null "field_url" => "/11-klass/algebra/mektep-abylkasymova1-uchebnik-2019/02-1" "field_display_title" => "2.1" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1203} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1212} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1208} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1206} "content" => Illuminate\Database\Eloquent\Collection {#1207} "next" => array:2 [ "refs" => "1124854" "type" => "task" ] "previous" => array:2 [ "refs" => "1124852" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1222} "page" => array:2 [ "refs" => "1100000" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Task {#1217 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1124854 "created_at" => "2026-04-10 13:58:26" "updated_at" => null 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"content" => Illuminate\Database\Eloquent\Collection {#1219 #items: array:2 [ 0 => App\Models\Element {#1237 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ …7] #original: array:7 [ …7] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 1 => App\Models\Element {#1239 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] 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#attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] }
№2.4 (с. 22)
Условие. №2.4 (с. 22)
скриншот условия
2.4. Вычислите площади криволинейных трапеций, изображенных на рисунке 14:
1)
$y = x^2 - 2x + 1$
2)
$y = 2 + 2x - x^2$
3)
$y = x^2$
4)
$y = 3\sin x$
Рис. 14
Решение 2 (rus). №2.4 (с. 22)
1)
Площадь криволинейной трапеции, ограниченной графиком функции $y=f(x)$, осью $Ox$ и прямыми $x=a$ и $x=b$, вычисляется по формуле Ньютона-Лейбница: $S = \int_a^b f(x)dx$.
В данном случае функция $f(x) = x^2 - 2x + 1$. Из графика видно, что пределы интегрирования: $a=0$ и $b=1$.
Вычислим определенный интеграл:
$S = \int_0^1 (x^2 - 2x + 1)dx$
Найдем первообразную для функции $f(x) = x^2 - 2x + 1$:
$F(x) = \frac{x^3}{3} - 2\frac{x^2}{2} + x = \frac{x^3}{3} - x^2 + x$
Теперь применим формулу Ньютона-Лейбница:
$S = F(1) - F(0) = \left. \left(\frac{x^3}{3} - x^2 + x \right) \right|_0^1 = \left(\frac{1^3}{3} - 1^2 + 1 \right) - \left(\frac{0^3}{3} - 0^2 + 0 \right) = \left(\frac{1}{3} - 1 + 1 \right) - 0 = \frac{1}{3}$
Ответ: $\frac{1}{3}$
2)
Функция, ограничивающая трапецию сверху: $y = 2 + 2x - x^2$. Пределы интегрирования по оси $Ox$, согласно рисунку, от $a=0$ до $b=2$.
Вычислим площадь с помощью определенного интеграла:
$S = \int_0^2 (2 + 2x - x^2)dx$
Находим первообразную:
$F(x) = 2x + 2\frac{x^2}{2} - \frac{x^3}{3} = 2x + x^2 - \frac{x^3}{3}$
Вычисляем площадь:
$S = F(2) - F(0) = \left. \left(2x + x^2 - \frac{x^3}{3} \right) \right|_0^2 = \left(2 \cdot 2 + 2^2 - \frac{2^3}{3} \right) - \left(2 \cdot 0 + 0^2 - \frac{0^3}{3} \right) = \left(4 + 4 - \frac{8}{3} \right) - 0 = 8 - \frac{8}{3} = \frac{24 - 8}{3} = \frac{16}{3}$
Ответ: $\frac{16}{3}$
3)
Фигура ограничена графиком функции $y = \frac{1}{4}x^4$ и прямыми $x=-2$ и $x=2$.
Функция $f(x) = \frac{1}{4}x^4$ является четной, так как $f(-x) = \frac{1}{4}(-x)^4 = \frac{1}{4}x^4 = f(x)$. Интервал интегрирования $[-2, 2]$ симметричен относительно нуля. Поэтому площадь можно вычислить как удвоенную площадь на отрезке $[0, 2]$:
$S = \int_{-2}^2 \frac{1}{4}x^4 dx = 2 \int_0^2 \frac{1}{4}x^4 dx = \frac{1}{2} \int_0^2 x^4 dx$
Находим первообразную для $x^4$:
$F(x) = \frac{x^5}{5}$
Вычисляем площадь:
$S = \frac{1}{2} \left. \left(\frac{x^5}{5} \right) \right|_0^2 = \frac{1}{2} \left(\frac{2^5}{5} - \frac{0^5}{5} \right) = \frac{1}{2} \left(\frac{32}{5} - 0 \right) = \frac{32}{10} = \frac{16}{5}$
Ответ: $\frac{16}{5}$
4)
Фигура ограничена графиком функции $y = 3\sin x$ и прямыми $x=\frac{\pi}{4}$ и $x=\pi$.
Вычисляем площадь с помощью определенного интеграла:
$S = \int_{\pi/4}^{\pi} 3\sin x dx$
Находим первообразную для $3\sin x$:
$F(x) = -3\cos x$
Вычисляем площадь:
$S = \left. (-3\cos x) \right|_{\pi/4}^{\pi} = (-3\cos(\pi)) - (-3\cos(\frac{\pi}{4})) = -3(-1) + 3\left(\frac{\sqrt{2}}{2}\right) = 3 + \frac{3\sqrt{2}}{2}$
Ответ: $3 + \frac{3\sqrt{2}}{2}$
Другие задания:
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