Question

State only the general formula for change of variables in double integrals. Explain the notation used.

Answer 1

In essence this is taking an integral in terms of  

x

’s and changing it into terms of  

u

’s. We want to do something similar for double and triple integrals. In fact we’ve already done this to a certain extent when we converted double integrals to polar coordinates and when we converted triple integrals to cylindrical or spherical coordinates. The main difference is that we didn’t actually go through the details of where the formulas came from. If you recall, in each of those cases we commented that we would justify the formulas for  

d

A

and  

d

V

eventually. Now is the time to do that justification.

While often the reason for changing variables is to get us an integral that we can do with the new variables, another reason for changing variables is to convert the region into a nicer region to work with. When we were converting the polar, cylindrical or spherical coordinates we didn’t worry about this change since it was easy enough to determine the new limits based on the given region. That is not always the case however. So, before we move into changing variables with multiple integrals we first need to see how the region may change with a change of variables.

First, we need a little terminology/notation out of the way. We call the equations that define the change of variables a transformation. Also, we will typically start out with a region,  

R

, in  

x

y

-coordinates and transform it into a region in  

u

v

-coordinates.