Find the volume of the solid in $\Bbb R^3$

Question

I need to find the volume of the solid in $\Bbb R^3$. It is bounded by the following: $y=x^2$, $x=y^2$, $z=x+y+21$ and $z=0$.

I known that the volume is expressed as follows:

$$\iiint 1 \, dV$$

I have trouble finding the correct limits of integration for this problem.

Answer 1

The projection of the solid on the $xy$ plane is the region enclosed by the parabolas $y = x^2$ and $x = y^2$. These intersect at $(0,0)$ and $(1,1)$ and form the shape of a leaf. Then:

$0 \le x \le 1$ and $x^2 \le y \le \sqrt{x}$.

$z$ moves from $0$ towards the plane $x + y + 21 = z$. Therefore:

$$V = \int_0^1 \int_{x^2}^{\sqrt{x}} \int_0^{x + y + 21} dz dy dx$$