TITLE: Volume of $n$ dimensional ellipsoid
QUESTION [11 upvotes]: Let $c_1,c_2,...,c_n$ be positive constants. Consider the $n$ dimensional ellipsoid given by $\{(x_1,...,x_n)|\sum_{k=1}^n\frac{x_k^2}{c_k^2}<1\}$. Prove that it's $n$ dimensional volume is $\frac{\pi^{n/2}}{\Gamma(n/2+1)}\prod_{k=1}^nc_k$

I'm know from http://en.wikipedia.org/wiki/Volume_of_an_n-ball that the volume of the $n$ ball is exactly the above formula with $c_1=c_2=...=c_n=r$, which at least confirms the formula in this special case. It seems we can argue by scaling in each coordinates, but how to make this rigorous?

REPLY [6 votes]: You know the formula for a ball. Note that the ellipsoid is just the ball scaled by a linear transformation. This linear transformation is diagonal with elements $c_1,c_2,\dots, c_n$. It is a standard fact (proved in Folland's Real Analysis, for example), that if $V$ is a Lebesgue measurable set and $L$ is a linear transformation, then the measure of $L(V)$ is $m(V)\cdot \det(L)$. It is easy to see that the determinant of the linear transformation of interest is $\prod_1^n c_i$, and this gives your desired result.