How does a region of integration change when making a rotation change of variables?

Question

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.

Answer 1

Maybe you can put $$\int_{0}^{\infty}\int_{0}^{\infty}\dots \int_{0}^{\infty}f(x_1,x_2,\dots,x_p)dx_1dx_2\dots dx_p$$ $$ = \frac{1}{2^p} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\dots \int_{-\infty}^{\infty}f(|x_1|,|x_2|,\dots,|x_p|)dx_1dx_2\dots dx_p.$$ Then, if the $f$ functions are even, you can rotate away.