Help with volume calculation via change of variables

Question

I'm struggling with correctly understanding the change of variables theorem. I know that for real numbers $a_1,a_n$, the set $\Delta _n$ bounded by the planes $x_i=0$ and the hyperplane $\sum \frac {x_i}{a_i}$ is $\frac{\prod_ia_i}{n!}$.

Now I need to calculate the volume of the pyramid with vertices $0,v_1,\dots ,v_n$ with $v_n\in \mathbb R^n$. So let $T:\mathbf e_i\to v_i$. By changing variables we have $$\int_{T\Delta}1=|\det V|\int _\Delta 1\circ T$$ where $\Delta$ is the unit pyramid of the same dimension, and $V$ is the matrix whose columns are $v_i$.

I'm now supposed to use what I know from the first paragraph to solve the problem, but I don't see how. What should I do?

Answer 1

In the RHS, $1\circ T=1$ since $1$ is the constant function and not the identity. Thus the RHS is $|\det V|\cdot \mathrm{vol}(\Delta)$, which requires the first paragraph to simplify.