The particular question is: The question is as follows: Find the volume of the solid bounded by the surfaces $z=y^2$ and $z=2-x^2$.
I really struggle to set up the integration bounds for these types of problems, and was hoping anyone could provide good strategies to figure out the bounds in addition to help with this specific problem.
Hint: the projection of the intersection of surfaces is $$y^2 = z = 2 − x^2\implies x^2 + y^2 = 2,$$ i.e., a circle. In the domain D: $x^2 + y^2\le 2$, obviously we have $y^2\le 2 − x^2$ and the volume will be $$V = \iint_D(2 - x^2 -y^2)\,dxdy.$$ (double integral of ceiling $-$ floor)
Try now cylindrical coordinates.