Cavalieri's principle for a 3 dimensional sphere

Question

By using Cavalieri's principle (Measure Theory) i want to prove for $R \geq 0$ and $B_R=\{(x,y,z) \in \mathbb{R}^n : x^2+y^2+z^2 \leq R^2\}$ that

$\lambda_3(B_R) = \frac{4}{3}\pi R^3$.

I already proved for $K=\{(x,y)\in \mathbb{R}^2 : x^2+y^2 \leq R^2\}$ that

$\lambda_2(K)=\pi R^2$.

I know i can use this to prove $\lambda_3(B_R)$ but in which way?

Answer 1

Using Cavalieri's principle you need to calculate

• $\int_{-R}^R\lambda_2(K(z))\;dz$ where
• $\lambda_2(K(z))= \pi r^2(z) = \pi (R^2-z^2)$

Now, evaluating gives immediately $$\pi\int_{-R}^R(R^2-z^2)\;dz =\frac{4}{3}\pi R^3$$