I assume the volume you are trying to find is the set (where ).
The Jacobian theorem says,
Here is a differenciable vector function.
And is the determinant of the Frechet' derivative i.e. the Jacobian.
Of course, is the set
It is conventient to have be such a function so that .
If we let then .
This means, the function (thinking of vectors in as coloums).
Now, this is convenient because
The volume of is given by,
Thus the volume is because the volume of a sphere is