Change of variables

Exercise

1.
Evaluate the following double integralsD
(a)
$\displaystyle{\iint_{\Omega}x^2dxdy, \ \Omega = \{(x,y) : x^2 + y^2 \leq 4\}}$
(b)
$\displaystyle{\iint_{\Omega}\log{(x^2+y^2)}dxdy, \ \Omega = \{(x,y) : 1 \leq x^2 + y^2 \leq 4\}}$
(c)
$\displaystyle{\iint_{\Omega}e^{(y-x)/(y+x)}dxdy, \ \Omega = \{(x,y) : x+y \leq 1, x \geq 0, y \geq 0 \}}$
(d)
$\displaystyle{\iint_{\Omega}e^{x^2 + y^2}dxdy, \ \Omega = \{(x,y) : 1 < x^2 + y^2 < 4 \}}$
(e)
$\displaystyle{\iint_{\Omega}\sqrt{1 - x^2 - y^2}dxdy, \ \Omega = \{(x,y) : x^2 + y^2 \leq 1 \}}$
(f)
$\displaystyle{\iint_{\Omega}(1 - x - 2y)dxdy, \ \Omega = \{(x,y) : x \geq 0, y \geq 0, x^2 + y^2 \leq 1 \}}$
2.
Using $u = x + y, v = x - y$, evaluate the following double integral.

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