We have seen how to evaluate a double integral
as an iterated integral of the form
in rectangular coordinates, because we know that
in rectangular coordinates. To make the change to polar coordinates, we not only need to represent the variables
and
in polar coordinates, but we also must understand how to write the area element,
in polar coordinates. That is, we must determine how the area element
can be written in terms of
and
in the context of polar coordinates. We address this question in the following activity.
From the result of
Activity 3.5.3, we see when we convert an integral from rectangular coordinates to polar coordinates, we must not only convert
and
to being in terms of
and
but we also have to change the area element to
in polar coordinates. As we saw in
Activity 3.5.3, the reason the additional factor of
in the polar area element is due to the fact that in polar coordinates, the cross sectional area element increases as
increases, while the cross sectional area element in rectangular coordinates is constant. So, given a double integral
in rectangular coordinates, to write a corresponding iterated integral in polar coordinates, we replace
with
with
and
with
Of course, we need to describe the region
in polar coordinates as well. To summarize:
While there is no firm rule for when polar coordinates can or should be used, they are a natural alternative anytime the domain of integration may be expressed simply in polar form, and/or when the integrand involves expressions such as