Integration Using Change of Variables - The Jacobian
Suppose thatandare functions continuously differentiable on a regionAsranges overthe pointgenerates a regionin theplane.
If the mappingis one to one on the interior ofand the Jacobianon the interior of B then the area of
Suppose then that we want to integrate some continuous functionoverIf the integral is intractable then we can change variables toand integrate overinstead, because
Proof: Break upintosmaller regionsWe can write
The last expression is a Riemann sum forand the expression tends to this integral as the diameter of thetends to zero.