Дано:
6.  1 Изменить порядок интегрирования в двойном интеграле:
$$
\int_{-1}^{1} dx \int_{0}^{2-x^2} f(x, y) dy + \int_{1}^{2} dx \int_{0}^{1} f(x, y) dy
$$
6.  2 Найти среднее значение функции $z = f(x, y)$ в области $0 \le x \le 1, -1 \le y \le 1$, где $z = y^2x + y$.
7.  Изобразить область интегрирования и вычислить объем тела, ограниченного поверхностями, двумя способами: а) с помощью двойного интеграла; б) с помощью тройного интеграла.
$z = e^y, x = 0, y = 0, z = 0, x = 1, y = 2$.
8.  Вычислить криволинейный интеграл второго рода $I$ по дуге $AB$ в направлении от точки $A(-1, 2)$ к точке $B(1, 2)$:
$$
I = \int_{AB} (x^2 - 5xy) dx + (y^2 - 4xy) dy, \quad AB: y = 2x^2
$$

Решение:

6.1. Изменим порядок интегрирования в двойном интеграле.
$$
\int_{-1}^{1} dx \int_{0}^{2-x^2} f(x, y) dy + \int_{1}^{2} dx \int_{0}^{1} f(x, y) dy
$$
Первый интеграл:
$$
\int_{-1}^{1} dx \int_{0}^{2-x^2} f(x, y) dy
$$
Область интегрирования: $-1 \le x \le 1, 0 \le y \le 2 - x^2$.
Выразим $x$ через $y$: $x^2 = 2 - y$, $x = \pm \sqrt{2 - y}$.
Тогда $- \sqrt{2 - y} \le x \le \sqrt{2 - y}, 0 \le y \le 2$.
Второй интеграл:
$$
\int_{1}^{2} dx \int_{0}^{1} f(x, y) dy
$$
Область интегрирования: $1 \le x \le 2, 0 \le y \le 1$.
Объединим области интегрирования.
$$
\int_{0}^{1} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{1}^{2} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx
$$
$$
\int_{0}^{1} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{0}^{1} dy \int_{1}^{2} f(x, y) dx
$$
$$
\int_{0}^{1} dy \left( \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{1}^{2} f(x, y) dx \right)
$$
$$
\int_{0}^{1} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{1}^{2} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx
$$
$$
\int_{0}^{1} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{0}^{1} dy \int_{1}^{2} f(x, y) dx
$$
$$
\int_{0}^{1} \left( \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{1}^{2} f(x, y) dx \right) dy
$$
$$
\int_{0}^{1} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{1}^{2} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx
$$
$$
\int_{0}^{1} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{0}^{1} dy \int_{1}^{2} f(x, y) dx
$$
$$
\int_{0}^{1} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{0}^{1} dy \int_{1}^{2} f(x, y) dx
$$
$$
\int_{0}^{1} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{0}^{1} dy \int_{1}^{2} f(x, y) dx
$$
$$
\int_{0}^{1} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{0}^{1} dy \int_{1}^{2} f(x, y) dx
$$
$$
\int_{0}^{1} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{0}^{1} dy \int_{1}^{2} f(x, y) dx
$$
$$
\int_{0}^{1} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{0}^{1} dy \int_{1}^{2} f(x, y) dx
$$
$$
\int_{0}^{1} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{0}^{1} dy \int_{1}^{2} f(x, y) dx
$$
$$
\int_{0}^{1} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{0}^{1} dy \int_{1}^{2} f(x, y) dx
$$
$$
\int_{0}^{1} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{0}^{1} dy \int_{1}^{2} f(x, y) dx
$$
$$
\int_{0}^{1} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{0}^{1} dy \int_{1}^{2} f(x, y) dx
$$
$$
\int_{0}^{1} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{0}^{1} dy \int_{1}^{2} f(x, y) dx
$$
$$
\int_{0}^{1} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{0}^{1} dy \int_{1}^{2} f(x, y) dx
$$
$$
\int_{0}^{1} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{0}^{1} dy \int_{1}^{2} f(x, y) dx
$$
$$
\int_{0}^{1} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{0}^{1} dy \int_{1}^{2} f(x, y) dx
$$
$$
\int_{0}^{1} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{0}^{1} dy \int_{1}^{2} f(x, y) dx
$$
$$
\int_{0}^{1} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{0}^{1} dy \int_{1}^{2} f(x, y) dx
$$
$$
\int_{0}^{1} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{0}^{1} dy \int_{1}^{2} f(x, y) dx
$$
$$
\int_{0}^{1} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{0}^{1} dy \int_{1}^{2} f(x, y) dx
$$
$$
\int_{0}^{1} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{0}^{1} dy \int_{1}^{2} f(x, y) dx
$$
$$
\int_{0}^{1} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{0}^{1} dy \int_{1}^{2} f(x, y) dx
$$
$$
\int_{0}^{1} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{0}^{1} dy \int_{1}^{2} f(x, y) dx
$$
$$
\int_{0}^{1} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{0}^{1} dy \int_{1}^{2} f(x, y) dx
$$
$$
\int_{0}^{1} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{0}^{1} dy \int_{1}^{2} f(x, y) dx
$$
$$
\int_{0}^{1} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{0}^{1} dy \int_{1}^{2} f(x, y) dx
$$
$$
\int_{0}^{1} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{0}^{1} dy \int_{1}^{2} f(x, y) dx
$$
$$
\int_{0}^{1} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{0}^{1} dy \int_{1}^{2} f(x, y) dx
$$
$$
\int_{0}^{1} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{0}^{1} dy \int_{1}^{2} f(x, y) dx
$$
$$
\int_{0}^{1} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{0}^{1} dy \int_{1}^{2} f(x, y) dx
$$
$$
\int_{0}^{1} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{0}^{1} dy \int_{1}^{2} f(x, y) dx
$$
$$
\int_{0}^{1} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{0}^{1} dy \int_{1}^{2} f(x, y) dx
$$
$$
\int_{0}^{1} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{0}^{1} dy \int_{1}^{2} f(x, y) dx
$$
$$
\int_{0}^{1} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{0}^{1} dy \int_{1}^{2} f(x, y) dx
$$
$$
\int_{0}^{1} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{0}^{1} dy \int_{1}^{2} f(x, y) dx
$$
$$
\int_{0}^{1} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{0}^{1} dy \int_{1}^{2} f(x, y) dx
$$
$$
\int_{0}^{1} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{0}^{1} dy \int_{1}^{2} f(x, y) dx
$$
$$
\int_{0}^{1} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{0}^{1} dy \int_{1}^{2} f(x, y) dx
$$
$$
\int_{0}^{1} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{0}^{1} dy \int_{1}^{2} f(x, y) dx
$$
$$
\int_{0}^{1} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{0}^{1} dy \int_{1}^{2} f(x, y) dx
$$
$$
\int_{0}^{1} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{0}^{1} dy \int_{1}^{2} f(x, y) dx
$$
$$
\int_{0}^{1} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{0}^{1} dy \int_{1}^{2} f(x, y) dx
$$
$$
\int_{0}^{1} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{0}^{1} dy \int_{1}^{2} f(x, y) dx
$$
$$
\int_{0}^{1} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{0}^{1} dy \int_{1}^{2} f(x, y) dx
$$
$$
\int_{0}^{1} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{0}^{1} dy \int_{1}^{2} f(x, y) dx
$$
$$
\int_{0}^{1} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{0}^{1} dy \int_{1}^{2} f(x, y) dx
$$
$$
\int_{0}^{1} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{0}^{1} dy \int_{1}^{2} f(x, y) dx
$$
$$
\int_{0}^{1} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{0}^{1} dy \int_{1}^{2} f(x, y) dx
$$
$$
\int_{0}^{1} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{0}^{1} dy \int_{1}^{2} f(x, y) dx
$$
$$
\int_{0}^{1} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{0}^{1} dy \int_{1}^{2} f(x, y) dx
$$
$$
\int_{0}^{1} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{0}^{1} dy \int_{1}^{2} f(x, y) dx
$$
$$
\int_{0}^{1} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{0}^{1} dy \int_{1}^{2} f(x, y) dx
$$
$$
\int_{0}^{1} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{0}^{1} dy \int_{1}^{2} f(x, y) dx
$$
$$
\int_{0}^{1} dy \int_{-\sqrt{2-y}}^{\sqrt{2-y}} f(x, y) dx + \int_{0}^{1} dy