Find the volume blocked between $z^2=x^2+y^2$ and $z=x^2+y^2$.
So one way is to use cylindrical coordinates, which actually worked for me, but I'm wondering why this solution doesn't work:
Our shapes intersect at $(0,0,0)$ and the plane $z=1$.
So we want to find the volume of: $\Omega = \left \{ (x,y,z) : z^2 \leq x^2+y^2 \leq z, 0 \leq z \leq 1\right \}$.
Using Fubini's Theorem: $$V = \int_{\Omega}1dV = \int_0^1 \int_{z^2 \leq x^2+y^2 \leq z} 1 dV^2dz$$
Now we can notice the inner integral is the area between 2 circles, so we have:
$$\int_0^1 \int_{z^2 \leq x^2+y^2 \leq z} 1 dV^2dz = \int_0^1 \pi z^2-\pi z^4dz= \frac{2 \pi}{15}$$
But the answer should be $$\frac{2 \pi}{12}$$
Where am I wrong?
Best Answer
$\displaystyle V = \int_0^1 (\pi z^2-\pi z^4dz) \ $ step has a mistake.
Please note that the area is $(\pi z - \pi z^2)$ and not $(\pi z^2-\pi z^4)$.
So the integral should be $V = \displaystyle \int_0^1 \pi (z - z^2) \ dz = \frac{\pi}{6}$