Evaluating a polar double integral on the semi disc

Question

The integral:

$$\iint_D (x^2-y^2)\,dx\,dy$$

where $D$ is defined as:

$$\{(x,y)\in \mathbb R^2 \mid x^2+y^2\le 1, x\ge 0\}$$

Context

I have solved double integrals on quarter discs but this semi disc is giving me a head ache.

Answer 1

So in polar coordinates it would be defined as:

$$0 \le r \le 1, 0 \le \theta \le \pi$$

So your double integral would be:

$$\int_0^{\pi}\int_0^1 r\cos(2\theta)rdrd\theta$$

Can you solve from here?

Answer 2

By symmetry the integral is $0$.

Remark: If you consider quarter-disks familiar, the half-disk can be broken up into two quarter-disks.