exo7 4381

Calculer \(\iint_D^{} f(x,y)\,d xd y\) :

1

\(D=\{y\ge0, x+y\le1, y-x\le1\}\),

\(f(x,y)=x^2y\).

2

\(D=\{x^2+y^2\le R^2\}\),

\(f(x,y)=x^2y\).

3

\(D=\{\frac{x^2}{a^2} + \frac{y^2}{b^2} \le 1\}\),

\(f(x,y)=x^2 + y^2\).

4

\(D=\{0 \le x \le 1-\frac{y^2}4\}\),

\(f(x,y)=x^2 + y^2\).

5

\(D=\{x^2+y^2 \le 1\}\),

\(f(x,y)=(x+y)^2\).

6

\(D=\{x^2+y^2 \le 1\}\),

\(f(x,y)=\frac{(x+y)^2}{x^2+y^2+1}\).

7

\(D=\{x\ge0, y\ge0, x+y\le1\}\),

\(f(x,y)=x+y+1\).

8

\(D=\{|x+y|\le1,|x-y|\le1\}\),

\(f(x,y)=\ln(x+y+1)\).

9

\(D=\{x\ge0, y\ge0, x+y\le\pi\}\),

\(f(x,y)=(x+y)\sin x\sin y\).

10

\(D=\{|x|\le x^2+y^2\le 1\}\),

\(f(x,y)=(1+x^2+y^2)^2\).

11

\(D=\{x\ge0, y\ge0, x+y\le a\}\),

\(f(x,y)=x+y+\sqrt{a^2+(x+y)^2}\).

12

\(D=\{x\ge0, y\ge0, x^2+y^2\le 1\}\),

\(f(x,y)=xy\sqrt{x^2+4y^2}\).

13

\(D=\{x^2+y^2-2y \le 0\}\),

\(f(x,y)=y\exp(x^2+y^2-2y)\).

14

\(D=\{y^2 \le 2px, x^2 \le 2py\}\),

\(f(x,y)=\exp\left(\frac{x^3+y^3}{xy}\right)\).