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Make a change of coordinates:
The Jacobian for this transformation is
and has a determinant of
Note that we need to use the Jacobian in the other direction; that is, we've computed
but we need the Jacobian determinant for the reverse transformation (from
to 
. To do this, notice that
we need to take the reciprocal of the Jacobian above.
The integral then changes to
The Jacobian for this transformation is
and has a determinant of
Note that we need to use the Jacobian in the other direction; that is, we've computed
but we need the Jacobian determinant for the reverse transformation (from
we need to take the reciprocal of the Jacobian above.
The integral then changes to