Computing the Jacobian for the change of variables from cartesian into spherical coordinates

Question

This is the question:

My question is whether the answer is $\rho^2\sin\phi$ or if it is $-\rho^2\sin\phi$ or if it doesn't necessarily matter, and why not.

I found a solution online that set up the determinant in the same way that I set up my determinant, however they got $\rho^2\sin\phi$ instead of $-\rho^2\sin\phi$ (which is what I got).

This is the solution online:

But I found another solution online that also set up the determinant the way that I set it up and it got $-\rho^2\sin\phi$ (which is what I got also).

This solution is:

So I just want to know which is the correct answer and why.

This is my solution:

Answer 1

If the Jacobian is negative, then it means that the orientation of the region of integration is flipped.

You must take the absolute value always.