My task is this;
Calculate$$\iint\limits_{A}y\:dA.$$
Where $A$ is the region in the $xy-$plane such that $x^2\leq y,\: x^2 + y^2 \leq 2$.
My work so far:
Our region $A$ is in the first and seccond quadrant above the parabola $x^2$ and below the circle centered at the origin with a radius of $\sqrt{2}$. Switching to polar coordinates gives us (remember the jacobian):$$\int\limits_{0}^{\pi}\int\limits_{r^2\cos^2(\theta)}^{\sqrt{2}}r^2\sin(\theta)\: dr\:d\theta.$$
However this setup leads to an answer with a variable $r$ and since the answer is a real number i must have set this one up wrong. Hints are welcome, and don't show calculations that reveal the answer as i would very much like to do that on my own:)
Thanks in advance!
Best Answer
Stick with rectangular coordinates. I get
$$\int_{-1}^1 dx \, \int_{x^2}^{\sqrt{2-x^2}} dy \, y$$
which I imagine you should be able to do easily.