We integrate over regions in polar coordinates.

We are currently interested in computing integrals of functions over various regions in and via Some regions like rectangles and boxes are easy to describe using -coordinates (a.k.a. rectangular coordinates). However, other regions like circles and other things with rotational symmetry are easier to work with in polar coordinates. Recall that in polar coordinates, where is a function of . When working with parametric equations of this form, it is common to notate and state that we are working in polar coordinates.
Consider the point in polar coordinates. What is this point when expressed in -coordinates?
Consider the point in -coordinates. What is this point when expressed in polar coordinates with ?

Double integrals in polar coordinates

The basic form of the double integral is:

which can be interpreted as

Over some region, sum up products of heights and areas.

Of course if you want to evaluate the integral (and honestly, who doesn’t?) you have to change to a region defined in -coordinates, and change to or leaving iterated integrals

Now consider representing a region with polar coordinates.

Let be the region in the first quadrant bounded by the curve. We can approximate this region using the natural shape of polar coordinates: Portions of sectors of circles. In the figure, one such region is shaded, shown below:

From the picture above, we see that:

So to evaluate replace with and convert the function to a function of polar coordinates: Finally, find bounds and that describe . Let’s state this as a theorem:

Write down a double integral in polar coordinates that will compute the area of a circle of radius .

Finally, let’s derive the volume of a sphere using a double integral in polar coordinates.

One may wonder how polar coordinates could be extended to triple integrals…read on!