How to find the volume of a solid using integration bound by 4 equations?

Question

The prompt is to find the volume of the solid bound by $z = x^2 + y^2$, $y = x^2$, $y = 1$ and $z = 0$.

Here's what it looks like after being plotted

I am completely lost on how to find the limits and whether I should use triple or double integration?

Answer 1

The projection of your solid on the $xy$ plane is a parabolic segment and by symmetry

$$ V=2\int_{0}^{1}\int_{x^2}^{1}\int_{0}^{x^2+y^2}1\,dz\,dy\,dx =2\int_{0}^{1}\int_{x^2}^{1}x^2+y^2\,dy\,dx\tag{1}$$ $$ V = 2\int_{0}^{1}\left(\frac{1}{3}+x^2-x^4-\frac{x^6}{3}\right)\,dx =2\left(\frac{1}{3}+\frac{1}{3}-\frac{1}{5}-\frac{1}{21}\right)=\color{red}{\frac{88}{105}}.\tag{2}$$

Answer 2

If you use triple integration, integrate 1 with the following limits. $0 \le z \le x^2+y^2$, $x^2 \le y \le 1$, $-1 \le x \le 1$. Do you see why?