By transforming to polar co-ordinates, evaluate the following integral.

Question

By transforming to polar co-ordinates, evaluate the following integral:

Converting the Integrand is simple for me, but I am confused on how I can convert the limits - how can I visual this region?

Answer 1

https://www.desmos.com/calculator/kywphz3yj9

After analyzing the bounds, we can see that this is a section of a circle, bounded by $y=x$, $y=$ [circle of radius $a$], $x=0$, and $x=a/\sqrt2$. This transforms into $\pi/4\le\theta\le\pi/2$ from $x=0\ \&\ y=x$ stopped at $x=a/\sqrt2$. And then $r$ is easy because it's the region inside a circle of radius $a$: $0\le r\le a$. $$\int_{\pi/4}^{\pi/2}d\theta \cdot \int_{0}^{a} r^2\,dr$$