In an integration exercise, I need to find the volume of the region bounded by the paraboloid $z=x^2+y^2$ and below by the triangle enclosed by the lines $y=x$, $x=0$, and $x+y=2$ in the $xy$-plane.
Here is the illustration of $xy$-plane and the standard solution:
However, I got some troubles when I try to use the method of polar coordinate transformation to solve it.
$\left\{ \begin{array}{ll} x=r\cdot cos\theta\\ y=r\cdot sin\theta \end{array} \right.$ , where the Jacobian is $J=r$.
The integral should be
$$ \iint_R[(r\cdot cos\theta)^2+(r\cdot sin\theta)^2]\cdot|J|~drd\theta=\iint_R r^3~drd\theta $$
I think the limits of $\theta$ should be $\frac{\pi}{4}\sim\frac{\pi}{2}$, but I'm not sure what are the inner limits. Thanks for helps.