calculating the volume of a set

Question

How can I calculate $\int_M1dV$ for $M:=\{(x,y,z)\in\mathbb R^3:x^2+y^2+z^2\leq 4, x^2+y^2\geq 1\}$ ? I think I need a diffeomorphism to calculate this integral. What diffeomorphism should I use?

Answer 1

Working with cylinders in spherical coordinate system is not very convenient, so I will stick to the cylindrical coordinates. Note that in the cylindrical coordinate $(r,\theta,z)$, your first constraint says $$r^2 \le 4 - z^2 \iff r \le \sqrt{4-z^2}$$ and the second constraint says $r^2 \ge 1 \iff r > 1$ (since $r>0$ is dictated by the coordinates themselves).

So it sounds like your volume is just $$ V = \int_{\theta = 0}^{\theta = 2\pi} \int_{z = -2}^{z=2} \int_{r = 1}^{r=\sqrt{4-z^2}} r drdzd\theta $$ by symmetry in $z$...