Suppose I have a p-dimensional integral:
$\int_{0}^{\infty}\int_{0}^{\infty}\dots \int_{0}^{\infty}f(x_1,x_2,\dots,x_p)dx_1dx_2\dots dx_p$
And then I make a rotation + translation transform:
$W=A^{T}(X-b)$
Question: How will the region of integration $X>0$ change in the $W$ space?
Can assume $A$ is a matrix of eigenvectors of a real symmetric positive definite matrix if this makes the answer easier.