Volume of $M = \{(x, y, z) \in \mathbb R^3 : z^2 \le x^2 + y^2 \le r^2, z \ge 0\}$

Question

I am having a hard time solving this. Is $M$ 2D or 3D? Isn't it an infinite cylinder which starts at the $x,y$ axis and grows vertically with $z$? Or is it just a circular disk at $x,y$ axis?

We are supposed to calculate this with Cavalieri's principle...

Answer 1

You want to integrate $\int_{0}^{r} \int_{z}^{r} \int_{0}^{2\pi} s\; d\phi\; ds\; dz$. Why?

Geometrically, you get a cylinder of radius $r$ and height $r$ with a hollowed out inverted cone. So you can also readily do this with Cavalieri's Priniple.

So, $Vol(M) = Vol(cylinder) - Vol(cone) = \pi r^2h (1 - 1/3) = \frac{2\pi r^2h}{3}$. Now, $h=r$ and so $Vol(M)=\frac{2\pi r^3}{3}$. I refer use to Wiki on Cavalieri's Principle for more details; there is a similar problem with a sphere.