From Convex Optimization by Boyd & Vanderberge:
Let $$E = \{v : \|Av + b\|_2 \le 1\}$$ be the equation of a general ellipsoid for $A \in S^n_{++}$ and $b \in \Bbb R^n$, where $S^n_{++}$ means positive definite symmetric matrices of size $n$. Then, the volume of $E$ is proportional to $\det \left( A^{-1} \right)$.
Can someone explain or prove why this is true? I have no idea how to show this and this is mentioned in passing.
This is a consequence of the multidimensional change of variables formula which states that (under suitable hypotheses) the following formula holds:
For your case, choose $f = 1$ (the constant function), let $B = \{\xi \in \mathbb{R}^n : \lVert \xi \rVert_2 \leq 1\}$ be the unit ball in $\mathbb{R}^n$, and define $g: \mathbb{R}^n \to \mathbb{R^n}$ by \begin{equation} g(\xi) = A^{-1} \cdot(\xi - b) \end{equation} Then, you can easily verify that $E = g(B)$, so that by the formula above, \begin{align} \text{vol}(E) &= \int_E 1 \\ &= \int_{g(V)} 1 \\ &= \int_B (1 \circ g) \cdot |\det g'| \\ &= \int_B 1 \cdot |\det A^{-1}| \\ &= \left( \det A^{-1} \right) \cdot \text{vol}(B) \end{align} In the last line, I got rid of the absolute values because of positive-definiteness. So, the volume of $E$ is indeed proportional as claimed, with the constant of proportionality being the volume of the unit ball in $\mathbb{R}^n$.