Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

What is the relationship between the Lebesgue measure of a ball in $\mathbb{R}^n$ to the measure of a sphere? I've derived the measure of a sphere $S^{n-1}$ in $\mathbb{R}^n$ to be $\frac{2\pi^{n/2}}{\Gamma(n/2)}$, but I don't know how to relate the two. Please help and thanks in advance!

share|improve this question

1 Answer 1

up vote 2 down vote accepted

The area of a sphere of radius $r$ is the derivative with respect to $r$ of the volume of the ball of radius $r$. (Think about the volume of a thin film of thickness $dr$ on the surface of the ball!) Since that volume is proportional to $r^n$, it follows that the area of the unit sphere is $n$ times the volume of the unit ball.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.