$\int_{0}^{6} \int_{0}^{y} x dx dy$ where $x = r \cos \theta, y = r \sin \theta, dx dy = r dr d \theta$

Question

Given $x = r \cos \theta, y = r \sin \theta, dx dy = r dr d \theta$, how can I evaluate the following integral: $\int_{0}^{6} \int_{0}^{y} x dx dy$

Answer 1

First, substitute everything in for $x\,dx\,dy$. Next find the limits of integration in terms of $r$ and $\theta$. Personally, I recommend drawing a picture of the region of integration.