When the method of substitution is used to solve an integral of the form three parts of the integral are changed, the limits, the function and the infinitesimal . So if the substitution is of the form the limits, and , are found by solving and and the function is expressed in terms of as .
Figure 28 shows why the needs to be changed. While the is the same length for all , the change as changes. The rate at which they change is precisely . This gives the relation
Hence the transformed integral can be written as
Here the is playing the part of the Jacobian that we will define.
Another change of coordinates that you have seen is the transformations from cartesian coordinates
to polar coordinates
Recall that a double integral in polar coordinates is expressed as
We can see from Figure 29 that the area elements change in size as increases. The circumference of a circle of radius is , so the length of an arc spanned by an angle is . Hence the area elements in polar coordinates are approximated by rectangles of width and length . Thus under the transformation from cartesian to polar coordinates we have the relation
that is, plays the same role as . This is why the term appears in the integrand. Here is playing the part of the Jacobian.