Volume of cone inside of two spheres

Question

I have to find the volume of the solid in the region: $$\Omega=\{(x,y,z)\in\mathbb{R}^{3}\mid1\leq x^{2}+y^{2}+z^{2}\leq4,z^{2}\leq x^{2}+y^{2},z\leq0\}$$ I know that, in spherical spherical coordinates, $0\leq\theta\leq2\pi$. I also got: $0\leq\phi\leq5\pi/6$, but I don't know if this is right.

But, for me, the main problem is: what is the variation of $\rho$?

Answer 1

We have

• $1\le r \le 2$
• $\frac{3\pi}4\le \phi \le \pi$

therefore the set up should be

$$V=\int_0^{2\pi}\, d\theta \int_{\frac{3\pi}4}^{\pi} \, d\phi\int_1^2r^2\sin\phi\,dr$$