Volume of the sphere of radius 5

Problem

Compute the volume of the sphere of radius $5$ using the integral.

Solution

We slice the sphere into discs of radius $r(z) = \sqrt{25-z^2}$ and thickness $dz$, with $z\in[-5,5]$: $V = \int_{-5}^{5}\pi(25-z^2)\,dz = \pi\left[25z - \frac{z^3}{3}\right]_{-5}^{5} = \pi\left(2\cdot 125 - \frac{2\cdot 125}{3}\right) = \boxed{\dfrac{500\pi}{3}}.$ Consistent with the general formula $V = \tfrac43\pi R^3 = \tfrac43\pi\cdot 125 = \tfrac{500}{3}\pi$.

Links

Topics: Integral
Concepts: Definite integral · Volume by cross-sections
Skills: Compute · Integrate
Exercise types: Integral computation