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Номер 355, страница 188 - гдз по алгебре 10-11 класс учебник Колмогоров, Абрамов
Авторы: Колмогоров А. Н., Абрамов А. М., Дудницын Ю. П.
Тип: Учебник
Издательство: Просвещение
Год издания: 2008 - 2026
Цвет обложки: зелёный, чёрный
ISBN: 978-5-09-019513-3
Рекомендовано Министерством образования и науки Российской Федерации
Алгебра и начала математического анализа
Популярные ГДЗ в 10 классе
Глава 3. Первообразная и интеграл. Параграф 8. Интеграл - номер 355, страница 188.
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" => 104181 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "188" "field_page_end" => null "field_url" => "/10-klass/algebra/kolmogorov/355" "field_display_title" => "355" "field_outside_task" => null "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" => 103772 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Первообразная и интеграл" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1050 #items: array:1 [ 0 => App\Models\Term {#1047 #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 [ …6] "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 [ …6] "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" => "174" "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 {#1083 #items: array:1 [ 0 => App\Models\Book {#1049 #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" => 30 "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 {#1051 #items: array:1 [ 0 => App\Models\Term {#1048 #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 [ …6] "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 [ …6] "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 {#1052 #items: array:2 [ 0 => App\Models\Term {#1053 #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" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ …6] "field_translit" => "desjatyj" ] #original: array:6 [ "id" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ …6] "field_translit" => "desjatyj" ] #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 {#1054 #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 [ …6] "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 [ …6] "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 {#1055 #items: array:1 [ 0 => App\Models\Term {#1056 #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" => 5153 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Просвещение" "field_cases" => null "field_translit" => "prosveschenie" ] #original: array:6 [ "id" => 5153 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Просвещение" "field_cases" => null "field_translit" => "prosveschenie" ] #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 {#1057 #items: array:3 [ 0 => App\Models\Term {#1058 #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" => 4632 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Колмогоров" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Андрей" "field_patronymic" => "Николаевич" "field_surname_rp" => "Колмогорова" ] #original: array:12 [ "id" => 4632 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Колмогоров" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Андрей" "field_patronymic" => "Николаевич" "field_surname_rp" => "Колмогорова" ] #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 {#1059 #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" => 3650 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Абрамов" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Александр" "field_patronymic" => "Михайлович" "field_surname_rp" => null ] #original: array:12 [ "id" => 3650 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Абрамов" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Александр" "field_patronymic" => "Михайлович" "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 => "*" ] } 2 => App\Models\Term {#1060 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 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#hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1061 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1062 #items: array:1 [ 0 => App\Models\Term {#1063 #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 [ …6] "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 [ …6] "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 {#1064 #items: array:1 [ 0 => App\Models\Term {#1065 #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" => 9 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Россия" "field_cases" => array:6 [ …6] "field_translit" => "rossiya" ] #original: array:6 [ "id" => 9 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Россия" "field_cases" => array:6 [ …6] "field_translit" => "rossiya" ] #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 {#1066 #items: array:1 [ 0 => App\Models\Term {#1067 #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" => 15 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Москва" "field_cases" => array:6 [ …6] "field_translit" => "moskva" ] #original: array:6 [ "id" => 15 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Москва" "field_cases" => array:6 [ …6] "field_translit" => "moskva" ] #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 {#1068 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1069 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1070 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1071 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => "17" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1072 #items: array:1 [ 0 => App\Models\Term {#1073 #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" => 33 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "стереотипное" "field_cases" => array:6 [ …6] "field_translit" => "stereotipnoe" ] #original: array:6 [ "id" => 33 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "стереотипное" "field_cases" => array:6 [ …6] "field_translit" => "stereotipnoe" ] #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_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1074 #items: array:1 [ 0 => App\Models\Term {#1058} ] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2008" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 10-11 классов общеобразовательных учреждений" "field_allowed" => "Рекомендовано Министерством образования и науки Российской Федерации" "field_reserve_field" => "Алгебра и начала математического анализа" "field_link_to_source" => null "field_tasks_count" => "1202" "field_priority" => "2" "field_default_folder" => "/algebra_10/kolmogorov/" "field_isbn" => "978-5-09-019513-3" "field_cover" => array:1 [ 0 => "/media/algebra_10/kolmogorov/covers/cover1.webp?ts=1733838489" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/kolmogorov/covers/cover1.webp?ts=1733838489" "alt" => "" "width" => "680" "height" => "915" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1075 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1076 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1077 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/algebra/kolmogorov" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1078 #items: array:2 [ 0 => App\Models\Term {#1079 #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" => 47 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "зелёный" "field_cases" => array:6 [ …6] "field_translit" => "zelyonyy" ] #original: array:6 [ "id" => 47 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "зелёный" "field_cases" => array:6 [ …6] "field_translit" => "zelyonyy" ] #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 {#1080 #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" => 56 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "чёрный" "field_cases" => array:6 [ …6] "field_translit" => "chyornyy" ] #original: array:6 [ "id" => 56 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "чёрный" "field_cases" => array:6 [ …6] "field_translit" => "chyornyy" ] #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" => "<!--Текст в низу страницы--> <p> Выбирая в качестве онлайн-консультанта <strong>ГДЗ по алгебре 10-11 класс учебник Колмогоров, Абрамов, Дудницын (Просвещение)</strong>, каждый школьник может быть уверен, что справится со всеми учебными задачами текущего года. Помимо идеальной подготовки домашних заданий, подросток сможет самостоятельно осваивать новые темы, выполнять диагностику своих знаний и восполнять в них пробелы, повторять и закреплять изученный материал. При этом ребенку не понадобятся дополнительные консультации педагога. Со всеми вопросами и темами текущего учебного года он справится самостоятельно. </p> <h2> Что готовит школьникам алгебра в 10-11 классах </h2> <p> Алгебру можно назвать лидером среди всех учебных дисциплин по количеству изучаемых тем. В десятом и одиннадцатом классах все усложняется тем, что помимо новых разделов, ребятам предстоит огромная работа по повторению ранее пройденного материала, ведь в конце года всех их ждет сложнейшее испытание - итоговый экзамен по предмету. Но его результат будет зависеть от того, насколько хорошо ребята освоят следующие разделы: </p> <ol> <li>Тригонометрические функции.</li> <li>Производная и её применения.</li> <li>Первообразная и интеграл.</li> <li>Показательная и логарифмическая функции.</li> <li>Действительные числа.</li> <li>Функции. </ol> <p> Материал очень насыщенный и достаточно сложный. При этом недостаточно будет просто вызубрить массу формул и правил. Простое заучивание - это лишь часть работы. Главное научиться применять полученные знания на практике. Именно с этим у большинства ребят возникают проблемы. Работа вместе с <strong>решебником по алгебре за 10-11 классы авторов Колмогорова, Абрамова, Дудницына</strong> позволит преодолеть все препятствия в учебе. Детально расписанная в пособии информация поможет прекрасно понять материал и научиться применять теорию на практике. </p> <h2> Что представляет собой решебник к учебнику Колмогорова (Просвещение) </h2> <p> Верные ответы — это далеко не все, что содержится во вспомогательном пособии. Сборник с готовыми домашними заданиями структурно и по содержанию полностью копирует учебник под редакцией Колмогорова и включает: </p> <ul> <li>тематические параграфы с соответствующими им упражнениями;</li> <li>номера к разделу «Задачи на повторение»;</li> <li>задачи повышенной трудности.</li> </ul> <p> Каждое решение расписано максимально подробно, с выкладками по теории, графиками и выводами. В особо сложных вопросах авторы дают дополнительные комментарии. Материал изложен простым языком, доступным для понимания абсолютно всем старшеклассникам. Электронный формат существенно облегчает работу с пособием, позволяет выбрать удобный ритм для учебы, экономит время и силы ребенка. </p> <h2> Зачем использовать ГДЗ по алгебре </h2> <p> Даже в выпускном классе нельзя опускать руки, если вдруг освоение алгебры не задалось. Еще есть время поправить свои оценки и знания. Главное правильно распределить свое время, найти верного помощника и вернуть интерес к предмету. Решебник позволит все эти элементы совместить воедино для достижения блестящего результата. Предложенное пособие станет тем консультантом, который даст возможность подростку быстро и качественно выполнить домашнее задание. Это в свою очередь позволит вернуть интерес к предмету и, что немало важно, выкроить ценные часы на отдых. Главное - не прибегать к списыванию готовых решений, ведь простое копирование поможет лишь на время, однако на первой же проверочной в классе обман вскроется. 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Помимо идеальной подготовки домашних заданий, подросток сможет самостоятельно осваивать новые темы, выполнять диагностику своих знаний и восполнять в них пробелы, повторять и закреплять изученный материал. При этом ребенку не понадобятся дополнительные консультации педагога. Со всеми вопросами и темами текущего учебного года он справится самостоятельно. </p> <h2> Что готовит школьникам алгебра в 10-11 классах </h2> <p> Алгебру можно назвать лидером среди всех учебных дисциплин по количеству изучаемых тем. В десятом и одиннадцатом классах все усложняется тем, что помимо новых разделов, ребятам предстоит огромная работа по повторению ранее пройденного материала, ведь в конце года всех их ждет сложнейшее испытание - итоговый экзамен по предмету. 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array:6 [ 0 => App\Models\Element {#1104 #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" => 156426 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1093 #items: array:1 [ 0 => App\Models\Edition {#1102 #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" => 644 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1100 …2} "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" => "vadim" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1099 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1097 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1098 …2} "field_content_source" => null ] #original: array:21 [ "id" => 644 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1100 …2} "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" => "vadim" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1099 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1097 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1098 …2} "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" => "104181" "type" => "task" ] "text" => "<p>Вычислите площадь фигуры, ограниченной линиями (355—356).</p><p>355.—</p><p><strong>а)</strong> $y = (x + 2)^2$, $y = 0$, $x = 0;$</p><p><strong>б)</strong> $y = \frac{1}{(x+1)^2} + 1$, $y = 0$, $x = 0$, $x = 2;$</p><p><strong>в)</strong> $y = 2x - x^2$, $y = 0;$</p><p><strong>г)</strong> $y = -(x - 1)^3$, $y = 0$, $x = 0.$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "355-1.jpg" "alt" => null "width" => "1418" "height" => 366 "path" => "/media/algebra_10/kolmogorov/0-00/355-1.webp?ts=1733864931" ] ] ] #original: array:7 [ "id" => 156426 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1093} "task" => array:2 [ "refs" => "104181" "type" => "task" ] "text" => "<p>Вычислите площадь фигуры, ограниченной линиями (355—356).</p><p>355.—</p><p><strong>а)</strong> $y = (x + 2)^2$, $y = 0$, $x = 0;$</p><p><strong>б)</strong> $y = \frac{1}{(x+1)^2} + 1$, $y = 0$, $x = 0$, $x = 2;$</p><p><strong>в)</strong> $y = 2x - x^2$, $y = 0;$</p><p><strong>г)</strong> $y = -(x - 1)^3$, $y = 0$, $x = 0.$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "355-1.jpg" "alt" => null "width" => "1418" "height" => 366 "path" => "/media/algebra_10/kolmogorov/0-00/355-1.webp?ts=1733864931" ] ] ] #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 {#1095 #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" => 157142 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1082 #items: array:1 [ 0 => App\Models\Edition {#1096 #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" => 645 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 1" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1092 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "vadim" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1088 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1090 …2} "field_content_source" => null ] #original: array:21 [ "id" => 645 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 1" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1092 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "vadim" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1088 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1090 …2} "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" => "104181" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "355-1.png" "alt" => null "width" => "764" "height" => 950 "path" => "/media/algebra_10/kolmogorov/1-00/355-1.webp?ts=1733865123" ] 1 => array:5 [ "name" => "355-2.png" "alt" => null "width" => "757" "height" => 627 "path" => "/media/algebra_10/kolmogorov/1-00/355-2.webp?ts=1733865123" ] ] ] #original: array:6 [ "id" => 157142 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1082} "task" => array:2 [ "refs" => "104181" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "355-1.png" "alt" => null "width" => "764" "height" => 950 "path" => "/media/algebra_10/kolmogorov/1-00/355-1.webp?ts=1733865123" ] 1 => array:5 [ "name" => "355-2.png" "alt" => null "width" => "757" "height" => 627 "path" => "/media/algebra_10/kolmogorov/1-00/355-2.webp?ts=1733865123" ] ] ] #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\Element {#1086 #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" => 157798 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1122 #items: array:1 [ 0 => App\Models\Edition {#1113 #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" => 646 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1117 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "vadim" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1135 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1116 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1132 …2} "field_content_source" => null ] #original: array:21 [ "id" => 646 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1117 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "vadim" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1135 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1116 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1132 …2} "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" => "104181" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "355-1.jpg" "alt" => null "width" => "1118" "height" => 1037 "path" => "/media/algebra_10/kolmogorov/2-00/355-1.webp?ts=1733995835" ] 1 => array:5 [ "name" => "355-2.jpg" "alt" => null "width" => "1286" "height" => 818 "path" => "/media/algebra_10/kolmogorov/2-00/355-2.webp?ts=1733995835" ] ] ] #original: array:6 [ "id" => 157798 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1122} "task" => array:2 [ "refs" => "104181" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "355-1.jpg" "alt" => null "width" => "1118" "height" => 1037 "path" => "/media/algebra_10/kolmogorov/2-00/355-1.webp?ts=1733995835" ] 1 => array:5 [ "name" => "355-2.jpg" "alt" => null "width" => "1286" "height" => 818 "path" => "/media/algebra_10/kolmogorov/2-00/355-2.webp?ts=1733995835" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 3 => App\Models\Element {#1119 #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" => 168507 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1144 #items: array:1 [ 0 => App\Models\Edition {#1133 #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" => 647 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 3" "field_order" => "4" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1137 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "vadim" "field_edition_type" => "solution" "field_root_dir" => "3-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1138 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1140 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1141 …2} "field_content_source" => null ] #original: array:21 [ "id" => 647 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 3" "field_order" => "4" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1137 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "vadim" "field_edition_type" => "solution" "field_root_dir" => "3-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1138 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1140 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1141 …2} "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" => "104181" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "355-1.jpg" "alt" => null "width" => "1118" "height" => 1037 "path" => "/media/algebra_10/kolmogorov/3-00/355-1.webp?ts=1733995803" ] ] ] #original: array:6 [ "id" => 168507 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1144} "task" => array:2 [ "refs" => "104181" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "355-1.jpg" "alt" => null "width" => "1118" "height" => 1037 "path" => "/media/algebra_10/kolmogorov/3-00/355-1.webp?ts=1733995803" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 4 => App\Models\Element {#1085 #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" => 168009 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1136 #items: array:1 [ 0 => App\Models\Edition {#1084 #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" => 648 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 4" "field_order" => "5" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1034 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "vadim" "field_edition_type" => "solution" "field_root_dir" => "4-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1043 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1039 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1041 …2} "field_content_source" => null ] #original: array:21 [ "id" => 648 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 4" "field_order" => "5" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1034 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "vadim" "field_edition_type" => "solution" "field_root_dir" => "4-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1043 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1039 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1041 …2} "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" => "104181" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "355-1.jpg" "alt" => null "width" => "1416" "height" => 745 "path" => "/media/algebra_10/kolmogorov/4-00/355-1.webp?ts=1733992623" ] 1 => array:5 [ "name" => "355-2.jpg" "alt" => null "width" => "1416" "height" => 1191 "path" => "/media/algebra_10/kolmogorov/4-00/355-2.webp?ts=1733992623" ] ] ] #original: array:6 [ "id" => 168009 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1136} "task" => array:2 [ "refs" => "104181" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "355-1.jpg" "alt" => null "width" => "1416" "height" => 745 "path" => "/media/algebra_10/kolmogorov/4-00/355-1.webp?ts=1733992623" ] 1 => array:5 [ "name" => "355-2.jpg" "alt" => null "width" => "1416" "height" => 1191 "path" => "/media/algebra_10/kolmogorov/4-00/355-2.webp?ts=1733992623" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 5 => App\Models\Element {#1142 #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" => 1341824 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1152 #items: array:1 [ 0 => App\Models\Edition {#1143 #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" => 5114 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 5" "field_order" => "6" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1146 …2} "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" => "vadim" "field_edition_type" => "solution" "field_root_dir" => "5-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1147 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1148 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1149 …2} "field_content_source" => null ] #original: array:21 [ "id" => 5114 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 5" "field_order" => "6" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1146 …2} "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" => "vadim" "field_edition_type" => "solution" "field_root_dir" => "5-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1147 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1148 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1149 …2} "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" => "104181" "type" => "task" ] "text" => "<p><strong>а)</strong> Фигура ограничена линиями $y = (x+2)^2$, $y = 0$ (ось Ox) и $x = 0$ (ось Oy).</p><p>График функции $y = (x+2)^2$ — это парабола, ветви которой направлены вверх, а вершина находится в точке $(-2, 0)$. Эта точка является точкой касания с осью Ox. Таким образом, фигура, площадь которой нужно найти, заключена между параболой, осью Ox и осью Oy. Пределы интегрирования по оси $x$ — от точки касания $x = -2$ до прямой $x = 0$.</p><p>Площадь фигуры вычисляется с помощью определенного интеграла. Поскольку на отрезке $[-2, 0]$ функция $y = (x+2)^2$ неотрицательна, площадь равна:</p><p>$S = \int_{-2}^{0} (x+2)^2 dx$</p><p>Для вычисления интеграла можно раскрыть скобки или использовать замену переменной. Раскроем скобки:</p><p>$S = \int_{-2}^{0} (x^2 + 4x + 4) dx = \left[ \frac{x^3}{3} + \frac{4x^2}{2} + 4x \right]_{-2}^{0} = \left[ \frac{x^3}{3} + 2x^2 + 4x \right]_{-2}^{0}$</p><p>Применяя формулу Ньютона-Лейбница, получаем:</p><p>$S = \left(\frac{0^3}{3} + 2 \cdot 0^2 + 4 \cdot 0\right) - \left(\frac{(-2)^3}{3} + 2(-2)^2 + 4(-2)\right) = 0 - \left(-\frac{8}{3} + 8 - 8\right) = - \left(-\frac{8}{3}\right) = \frac{8}{3}$</p><p>Ответ: $S = \frac{8}{3}$</p><p><strong>б)</strong> Фигура ограничена линиями $y = \frac{1}{(x+1)^2} + 1$, $y = 0$ (ось Ox), $x = 0$ и $x = 2$.</p><p>Фигура представляет собой криволинейную трапецию, ограниченную сверху графиком функции $y = \frac{1}{(x+1)^2} + 1$, снизу — осью Ox, и с боков — прямыми $x=0$ и $x=2$. Пределы интегрирования заданы в условии: от $a=0$ до $b=2$. На этом промежутке функция $y$ положительна, так как $\frac{1}{(x+1)^2} > 0$.</p><p>Площадь фигуры вычисляется как определенный интеграл:</p><p>$S = \int_{0}^{2} \left(\frac{1}{(x+1)^2} + 1\right) dx = \int_{0}^{2} (x+1)^{-2} dx + \int_{0}^{2} 1 dx$</p><p>Найдем первообразную:</p><p>$\int \left((x+1)^{-2} + 1\right) dx = \frac{(x+1)^{-1}}{-1} + x = -\frac{1}{x+1} + x$</p><p>Подставим пределы интегрирования:</p><p>$S = \left[ -\frac{1}{x+1} + x \right]_{0}^{2} = \left(-\frac{1}{2+1} + 2\right) - \left(-\frac{1}{0+1} + 0\right) = \left(-\frac{1}{3} + 2\right) - (-1) = \frac{5}{3} + 1 = \frac{8}{3}$</p><p>Ответ: $S = \frac{8}{3}$</p><p><strong>в)</strong> Фигура ограничена линиями $y = 2x - x^2$ и $y = 0$ (ось Ox).</p><p>График функции $y = 2x - x^2$ — это парабола, ветви которой направлены вниз. Чтобы найти пределы интегрирования, найдем точки пересечения параболы с осью Ox, решив уравнение $y=0$:</p><p>$2x - x^2 = 0 \implies x(2-x) = 0$</p><p>Отсюда получаем $x_1 = 0$ и $x_2 = 2$. Это и будут пределы интегрирования. На интервале $(0, 2)$ функция $y=2x-x^2$ положительна (например, при $x=1$, $y=1$).</p><p>Площадь фигуры равна:</p><p>$S = \int_{0}^{2} (2x - x^2) dx$</p><p>Вычислим интеграл:</p><p>$S = \left[ 2\frac{x^2}{2} - \frac{x^3}{3} \right]_{0}^{2} = \left[ x^2 - \frac{x^3}{3} \right]_{0}^{2} = \left(2^2 - \frac{2^3}{3}\right) - \left(0^2 - \frac{0^3}{3}\right) = 4 - \frac{8}{3} = \frac{12-8}{3} = \frac{4}{3}$</p><p>Ответ: $S = \frac{4}{3}$</p><p><strong>г)</strong> Фигура ограничена линиями $y = -(x - 1)^3$, $y = 0$ (ось Ox) и $x = 0$ (ось Oy).</p><p>График функции $y = -(x - 1)^3$ — это кубическая парабола. Найдем точку ее пересечения с осью Ox:</p><p>$-(x - 1)^3 = 0 \implies x - 1 = 0 \implies x = 1$</p><p>Фигура ограничена кривой, осью Ox ($y=0$) и осью Oy ($x=0$). Таким образом, пределы интегрирования по оси $x$ — от $0$ до $1$.</p><p>На отрезке $[0, 1)$ значение $(x-1)$ отрицательно, следовательно $(x-1)^3$ отрицательно, а $y = -(x-1)^3$ положительно. Значит, график функции находится над осью Ox.</p><p>Площадь фигуры равна:</p><p>$S = \int_{0}^{1} -(x-1)^3 dx$</p><p>Вычислим интеграл, используя замену $u=x-1$, $du=dx$. Новые пределы: если $x=0$, то $u=-1$; если $x=1$, то $u=0$.</p><p>$S = \int_{-1}^{0} -u^3 du = \left[ -\frac{u^4}{4} \right]_{-1}^{0} = \left(-\frac{0^4}{4}\right) - \left(-\frac{(-1)^4}{4}\right) = 0 - \left(-\frac{1}{4}\right) = \frac{1}{4}$</p><p>Ответ: $S = \frac{1}{4}$</p>" ] #original: array:6 [ "id" => 1341824 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1152} "task" => array:2 [ "refs" => "104181" "type" => "task" ] "text" => "<p><strong>а)</strong> Фигура ограничена линиями $y = (x+2)^2$, $y = 0$ (ось Ox) и $x = 0$ (ось Oy).</p><p>График функции $y = (x+2)^2$ — это парабола, ветви которой направлены вверх, а вершина находится в точке $(-2, 0)$. Эта точка является точкой касания с осью Ox. Таким образом, фигура, площадь которой нужно найти, заключена между параболой, осью Ox и осью Oy. Пределы интегрирования по оси $x$ — от точки касания $x = -2$ до прямой $x = 0$.</p><p>Площадь фигуры вычисляется с помощью определенного интеграла. Поскольку на отрезке $[-2, 0]$ функция $y = (x+2)^2$ неотрицательна, площадь равна:</p><p>$S = \int_{-2}^{0} (x+2)^2 dx$</p><p>Для вычисления интеграла можно раскрыть скобки или использовать замену переменной. Раскроем скобки:</p><p>$S = \int_{-2}^{0} (x^2 + 4x + 4) dx = \left[ \frac{x^3}{3} + \frac{4x^2}{2} + 4x \right]_{-2}^{0} = \left[ \frac{x^3}{3} + 2x^2 + 4x \right]_{-2}^{0}$</p><p>Применяя формулу Ньютона-Лейбница, получаем:</p><p>$S = \left(\frac{0^3}{3} + 2 \cdot 0^2 + 4 \cdot 0\right) - \left(\frac{(-2)^3}{3} + 2(-2)^2 + 4(-2)\right) = 0 - \left(-\frac{8}{3} + 8 - 8\right) = - \left(-\frac{8}{3}\right) = \frac{8}{3}$</p><p>Ответ: $S = \frac{8}{3}$</p><p><strong>б)</strong> Фигура ограничена линиями $y = \frac{1}{(x+1)^2} + 1$, $y = 0$ (ось Ox), $x = 0$ и $x = 2$.</p><p>Фигура представляет собой криволинейную трапецию, ограниченную сверху графиком функции $y = \frac{1}{(x+1)^2} + 1$, снизу — осью Ox, и с боков — прямыми $x=0$ и $x=2$. Пределы интегрирования заданы в условии: от $a=0$ до $b=2$. На этом промежутке функция $y$ положительна, так как $\frac{1}{(x+1)^2} > 0$.</p><p>Площадь фигуры вычисляется как определенный интеграл:</p><p>$S = \int_{0}^{2} \left(\frac{1}{(x+1)^2} + 1\right) dx = \int_{0}^{2} (x+1)^{-2} dx + \int_{0}^{2} 1 dx$</p><p>Найдем первообразную:</p><p>$\int \left((x+1)^{-2} + 1\right) dx = \frac{(x+1)^{-1}}{-1} + x = -\frac{1}{x+1} + x$</p><p>Подставим пределы интегрирования:</p><p>$S = \left[ -\frac{1}{x+1} + x \right]_{0}^{2} = \left(-\frac{1}{2+1} + 2\right) - \left(-\frac{1}{0+1} + 0\right) = \left(-\frac{1}{3} + 2\right) - (-1) = \frac{5}{3} + 1 = \frac{8}{3}$</p><p>Ответ: $S = \frac{8}{3}$</p><p><strong>в)</strong> Фигура ограничена линиями $y = 2x - x^2$ и $y = 0$ (ось Ox).</p><p>График функции $y = 2x - x^2$ — это парабола, ветви которой направлены вниз. Чтобы найти пределы интегрирования, найдем точки пересечения параболы с осью Ox, решив уравнение $y=0$:</p><p>$2x - x^2 = 0 \implies x(2-x) = 0$</p><p>Отсюда получаем $x_1 = 0$ и $x_2 = 2$. Это и будут пределы интегрирования. На интервале $(0, 2)$ функция $y=2x-x^2$ положительна (например, при $x=1$, $y=1$).</p><p>Площадь фигуры равна:</p><p>$S = \int_{0}^{2} (2x - x^2) dx$</p><p>Вычислим интеграл:</p><p>$S = \left[ 2\frac{x^2}{2} - \frac{x^3}{3} \right]_{0}^{2} = \left[ x^2 - \frac{x^3}{3} \right]_{0}^{2} = \left(2^2 - \frac{2^3}{3}\right) - \left(0^2 - \frac{0^3}{3}\right) = 4 - \frac{8}{3} = \frac{12-8}{3} = \frac{4}{3}$</p><p>Ответ: $S = \frac{4}{3}$</p><p><strong>г)</strong> Фигура ограничена линиями $y = -(x - 1)^3$, $y = 0$ (ось Ox) и $x = 0$ (ось Oy).</p><p>График функции $y = -(x - 1)^3$ — это кубическая парабола. Найдем точку ее пересечения с осью Ox:</p><p>$-(x - 1)^3 = 0 \implies x - 1 = 0 \implies x = 1$</p><p>Фигура ограничена кривой, осью Ox ($y=0$) и осью Oy ($x=0$). Таким образом, пределы интегрирования по оси $x$ — от $0$ до $1$.</p><p>На отрезке $[0, 1)$ значение $(x-1)$ отрицательно, следовательно $(x-1)^3$ отрицательно, а $y = -(x-1)^3$ положительно. Значит, график функции находится над осью Ox.</p><p>Площадь фигуры равна:</p><p>$S = \int_{0}^{1} -(x-1)^3 dx$</p><p>Вычислим интеграл, используя замену $u=x-1$, $du=dx$. Новые пределы: если $x=0$, то $u=-1$; если $x=1$, то $u=0$.</p><p>$S = \int_{-1}^{0} -u^3 du = \left[ -\frac{u^4}{4} \right]_{-1}^{0} = \left(-\frac{0^4}{4}\right) - \left(-\frac{(-1)^4}{4}\right) = 0 - \left(-\frac{1}{4}\right) = \frac{1}{4}$</p><p>Ответ: $S = \frac{1}{4}$</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" => "104182" "type" => "task" ] "previous" => array:2 [ "refs" => "104180" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1089 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1107 #items: array:1 [ 0 => App\Models\BookPage {#1110 #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" => 863547 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "188" "field_url" => "/10-klass/algebra/kolmogorov/page-188" "field_display_title" => "188" "field_folder" => "1" "field_image_name" => "188" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1109 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1042 #items: array:1 [ 0 => App\Models\Book {#1049} ] #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 {#1111 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1159 #items: array:1 [ 0 => App\Models\Element {#1158 #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" => 1037725 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1160 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1037725 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1160 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #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" => "863548" "type" => "book_page" ] "previous" => array:2 [ "refs" => "863546" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1258 #items: array:4 [ 0 => App\Models\Task {#1268 #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" => 104179 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "188" "field_page_end" => null "field_url" => "/10-klass/algebra/kolmogorov/353" "field_display_title" => "353" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1269 …2} "field_metatags_title" => null "field_metatags_description" => null 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№355 (с. 188)
Условие. №355 (с. 188)
скриншот условия
Вычислите площадь фигуры, ограниченной линиями (355—356).
355.—
а) $y = (x + 2)^2$, $y = 0$, $x = 0;$
б) $y = \frac{1}{(x+1)^2} + 1$, $y = 0$, $x = 0$, $x = 2;$
в) $y = 2x - x^2$, $y = 0;$
г) $y = -(x - 1)^3$, $y = 0$, $x = 0.$
Решение 1. №355 (с. 188)
Решение 2. №355 (с. 188)
Решение 3. №355 (с. 188)
Решение 4. №355 (с. 188)
Решение 5. №355 (с. 188)
а) Фигура ограничена линиями $y = (x+2)^2$, $y = 0$ (ось Ox) и $x = 0$ (ось Oy).
График функции $y = (x+2)^2$ — это парабола, ветви которой направлены вверх, а вершина находится в точке $(-2, 0)$. Эта точка является точкой касания с осью Ox. Таким образом, фигура, площадь которой нужно найти, заключена между параболой, осью Ox и осью Oy. Пределы интегрирования по оси $x$ — от точки касания $x = -2$ до прямой $x = 0$.
Площадь фигуры вычисляется с помощью определенного интеграла. Поскольку на отрезке $[-2, 0]$ функция $y = (x+2)^2$ неотрицательна, площадь равна:
$S = \int_{-2}^{0} (x+2)^2 dx$
Для вычисления интеграла можно раскрыть скобки или использовать замену переменной. Раскроем скобки:
$S = \int_{-2}^{0} (x^2 + 4x + 4) dx = \left[ \frac{x^3}{3} + \frac{4x^2}{2} + 4x \right]_{-2}^{0} = \left[ \frac{x^3}{3} + 2x^2 + 4x \right]_{-2}^{0}$
Применяя формулу Ньютона-Лейбница, получаем:
$S = \left(\frac{0^3}{3} + 2 \cdot 0^2 + 4 \cdot 0\right) - \left(\frac{(-2)^3}{3} + 2(-2)^2 + 4(-2)\right) = 0 - \left(-\frac{8}{3} + 8 - 8\right) = - \left(-\frac{8}{3}\right) = \frac{8}{3}$
Ответ: $S = \frac{8}{3}$
б) Фигура ограничена линиями $y = \frac{1}{(x+1)^2} + 1$, $y = 0$ (ось Ox), $x = 0$ и $x = 2$.
Фигура представляет собой криволинейную трапецию, ограниченную сверху графиком функции $y = \frac{1}{(x+1)^2} + 1$, снизу — осью Ox, и с боков — прямыми $x=0$ и $x=2$. Пределы интегрирования заданы в условии: от $a=0$ до $b=2$. На этом промежутке функция $y$ положительна, так как $\frac{1}{(x+1)^2} > 0$.
Площадь фигуры вычисляется как определенный интеграл:
$S = \int_{0}^{2} \left(\frac{1}{(x+1)^2} + 1\right) dx = \int_{0}^{2} (x+1)^{-2} dx + \int_{0}^{2} 1 dx$
Найдем первообразную:
$\int \left((x+1)^{-2} + 1\right) dx = \frac{(x+1)^{-1}}{-1} + x = -\frac{1}{x+1} + x$
Подставим пределы интегрирования:
$S = \left[ -\frac{1}{x+1} + x \right]_{0}^{2} = \left(-\frac{1}{2+1} + 2\right) - \left(-\frac{1}{0+1} + 0\right) = \left(-\frac{1}{3} + 2\right) - (-1) = \frac{5}{3} + 1 = \frac{8}{3}$
Ответ: $S = \frac{8}{3}$
в) Фигура ограничена линиями $y = 2x - x^2$ и $y = 0$ (ось Ox).
График функции $y = 2x - x^2$ — это парабола, ветви которой направлены вниз. Чтобы найти пределы интегрирования, найдем точки пересечения параболы с осью Ox, решив уравнение $y=0$:
$2x - x^2 = 0 \implies x(2-x) = 0$
Отсюда получаем $x_1 = 0$ и $x_2 = 2$. Это и будут пределы интегрирования. На интервале $(0, 2)$ функция $y=2x-x^2$ положительна (например, при $x=1$, $y=1$).
Площадь фигуры равна:
$S = \int_{0}^{2} (2x - x^2) dx$
Вычислим интеграл:
$S = \left[ 2\frac{x^2}{2} - \frac{x^3}{3} \right]_{0}^{2} = \left[ x^2 - \frac{x^3}{3} \right]_{0}^{2} = \left(2^2 - \frac{2^3}{3}\right) - \left(0^2 - \frac{0^3}{3}\right) = 4 - \frac{8}{3} = \frac{12-8}{3} = \frac{4}{3}$
Ответ: $S = \frac{4}{3}$
г) Фигура ограничена линиями $y = -(x - 1)^3$, $y = 0$ (ось Ox) и $x = 0$ (ось Oy).
График функции $y = -(x - 1)^3$ — это кубическая парабола. Найдем точку ее пересечения с осью Ox:
$-(x - 1)^3 = 0 \implies x - 1 = 0 \implies x = 1$
Фигура ограничена кривой, осью Ox ($y=0$) и осью Oy ($x=0$). Таким образом, пределы интегрирования по оси $x$ — от $0$ до $1$.
На отрезке $[0, 1)$ значение $(x-1)$ отрицательно, следовательно $(x-1)^3$ отрицательно, а $y = -(x-1)^3$ положительно. Значит, график функции находится над осью Ox.
Площадь фигуры равна:
$S = \int_{0}^{1} -(x-1)^3 dx$
Вычислим интеграл, используя замену $u=x-1$, $du=dx$. Новые пределы: если $x=0$, то $u=-1$; если $x=1$, то $u=0$.
$S = \int_{-1}^{0} -u^3 du = \left[ -\frac{u^4}{4} \right]_{-1}^{0} = \left(-\frac{0^4}{4}\right) - \left(-\frac{(-1)^4}{4}\right) = 0 - \left(-\frac{1}{4}\right) = \frac{1}{4}$
Ответ: $S = \frac{1}{4}$
Другие задания:
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