Improper integrals

Exercise

1.

Find the following improper integrals．

(a)

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

(b)

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

(c)

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

(d)

$${\iint_{\Omega}\frac{1}{x^2 y^2}dxdy, \ \Omega = \{(x,y) : x \geq 1, y \geq 1 \}}$$

(e)

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

(f)

$${\iint_{\Omega}\frac{dxdy}{1 + (x^2 + y^2)^2}, \ \Omega = \{(x,y) : -\infty < x,y < \infty \}}$$

2.

Show that， $${\Gamma\left(\frac{1}{2}\right) = \int_{0}^{\infty} x^{-\frac{1}{2}}e^{-x} dx = \sqrt{\pi}}$$.