Tranforming to polar co-ordinates

Question

$$I = \int_0^1\int_0^{\sqrt{1-x^2}} xy \, dy\, dx$$

By transforming to circular polar co-ordinates, evaluate I.

How do I do this?

Is there a formula/strategy for doing this that works with different problems too?

Answer 1

First, draw the integration area, meaning the set $$\{(x,y)| x\in[0,1] \wedge y\in [0,\sqrt{1-x^2}]\}.$$

You will see that the area you draw is very simple. Once you have it, it will be easy to parametrize it using polar coordinates.