(a) By considering an appropriate double integral, show that

$$\int_{0}^{\infty} e^{-a x^{2}} d x=\sqrt{\frac{\pi}{4 a}}$$

where $a>0$.

(b) Calculate $\int_{0}^{1} x^{y} d y$, treating $x$ as a constant, and hence show that

$$\int_{0}^{\infty} \frac{\left(e^{-u}-e^{-2 u}\right)}{u} d u=\log 2$$

(c) Consider the region $\mathcal{D}$ in the $x-y$ plane enclosed by $x^{2}+y^{2}=4, y=1$, and $y=\sqrt{3} x$ with $1<y<\sqrt{3} x$.

Sketch $\mathcal{D}$, indicating any relevant polar angles.

A surface $\mathcal{S}$ is given by $z=x y /\left(x^{2}+y^{2}\right)$. Calculate the volume below this surface and above $\mathcal{D}$.