I have the following problem:
Find the volume of the region bounded above by the cylinder $$ z=x^2$$ and below by the region enclosed by the parabola $$y=2-x^2$$ and the line $$y=x$$
I'm really struggling to visualize this region and come up with the limits of integration. I've made a couple of failed attempts at sketching the graph, none of which made any sense. I can't seem to find the correct region that satisfies the requirements. Can a kind soul help me with this?
Hint. I guess you would like to evaluate the volume of the solid $$S:=\{(x,y,z)\in\mathbb{R}^3:0\leq z\leq x^2,x\leq y\leq 2-x^2,\;x\in[-2,1]\}.$$ Then $$\mbox{Volume(S)}=\int_{x=-2}^{1}\int_{y=x}^{2-x^2}\int_{z=0}^{x^2}dzdydx.$$