Change of variables

Exercise

1.

Evaluate the following double integrals．

(a)

$${\iint_{\Omega}x^2dxdy, \ \Omega = \{(x,y) : x^2 + y^2 \leq 4\}}$$

(b)

$${\iint_{\Omega}\log{(x^2+y^2)}dxdy, \ \Omega = \{(x,y) : 1 \leq x^2 + y^2 \leq 4\}}$$

(c)

$${\iint_{\Omega}e^{(y-x)/(y+x)}dxdy, \ \Omega = \{(x,y) : x+y \leq 1, x \geq 0, y \geq 0 \}}$$

(d)

$${\iint_{\Omega}e^{x^2 + y^2}dxdy, \ \Omega = \{(x,y) : 1 < x^2 + y^2 < 4 \}}$$

(e)

$${\iint_{\Omega}\sqrt{1 - x^2 - y^2}dxdy, \ \Omega = \{(x,y) : x^2 + y^2 \leq 1 \}}$$

(f)

$${\iint_{\Omega}(1 - x - 2y)dxdy, \ \Omega = \{(x,y) : x \geq 0, y \geq 0, x^2 + y^2 \leq 1 \}}$$

2.

Using $u = x + y, v = x - y$, evaluate the following double integral.

$${\iint_{\Omega}(x^2 + y^2) e^{-x+y} dx dy, \ \Omega = \{-1 \leq x+y \leq 1, -1 \leq x - y \leq 1 \}}$$