Issues calculating double intergral

Question

Calculate $$\iint_\Omega(x^2+y^2)^b\,dxdy$$ where $\Omega=\{(x,y)\in\mathbb R^2\,:\, x^2+y^2<a^2\}$ and $b>0$ is a real constant.

I'm having trouble calculating the following, it's from a past exam that I'm doing for revision . I think I need to convert to polar co-ordinates and got an answer but it was $\dfrac{a^{4b+4}}{2b+2}$

Answer 1

Answer:

$\int\int_{R} (x^2+y^2)^b dxdy$ in the Domain $x^2+y^2\le a^2$ with b>0

==>

$$\int_{0}^{2\pi}\int_{0}^{a} r^{2b}rdrd\theta$$

Which when you evaluate gives

you

$$=\dfrac{\pi a^{2b+2}}{b+1}$$

Answer 2

convert to polar - the angle integral gives a factor of $2\pi$ and don't forget that $dx\;dy \to rdr\; d\theta$

$$I = 2\pi \int_0^a r^{2b+1} dr =\frac{2\pi a^{2b+2}}{2b+2}=\frac \pi {b+1} a^{2b+2} $$