Let $B = \{(x, y, z): x^2 +y^2 +z^2 ≤ 4\}$ be the ball with radius $2$ in $\mathbb{R}^3$ and let $V$ be the region inside $B$ above the plane $z = 1$. Use Cavalieri’s Principle to compute the volume of $V$.
I understand how to use Cavalieri’s Principle as you simply find the area of the cross section and then integrate that between the height. However, I don't understand how to do it for only part of the sphere. Please help.
The region above the plane $z=1$ has height between $1$ and $2$ (the radius of the sphere). Thus, you only need to integrate
$$ I= \int_1^2 Area(x^2+y^2 \leq 4-h^2)dh $$ Note that the region $x^2+y^2 \leq 4-h^2$ is a circle of radius $\sqrt{4-h^2}$. Therefore the integral becomes $$ I = \int_1^2 \pi(4-h^2)dh$$