Evaluating double integral by change in variable.

Question

Evaluate the following integral by changing to polar coordinates:
$\iint x dxdy$, where $0\leq x \leq y$ and $0 \leq y \leq 1$.
Above integral can be evaluated directly without changing the variables. I am getting the answer $\frac{1}{6}$. But I have to evaluate it by changing in polar coordinates. So let $x=r\cos\theta$ and $y=r\sin\theta$. Then $dxdy=rdrd\theta$. But what will be the limits on $r$ and $\theta$ ?

Answer 1

Hint: Draw the region is key to understand the integral. If you do the graph you will find $$\frac{\pi}{4}\leq \theta\leq \frac{\pi}{2}$$ and $$0\leq y\leq 1$$ $$0\leq r \text{sin}(x)\leq 1$$ $$0\leq r\leq\frac{1}{\text{sin}(x)}$$