Area Between Two Polar Curves - Calculus 2

At the very beginning of Calculus 2, we learned how to find the area between two curves within the rectangular coordinate system by using integration. This process involved identifying a top and bottom curve for the area we wanted to find, as well as the two values of x that the area was located between. Then, using a definite integral we subtracted the area beneath the bottom curve from the area beneath the top curve to find the area between the two curves.

This same process can applied to the polar coordinate system.

To find the area between two polar curves, we need to identify which polar curve bounds the area above (bigger radius), and which polar curve bounds the area below (smaller radius). Once these curves are identified, we then need to determine the points where the curves intersect, as the angles of those points will be the angles that the area lies between and thus will be our bounds of integration. From there, we can find the area between the curves by subtracting the area below the bottom curve that has the smaller radius from the area below the top curve that has the bigger radius.

This process should sound very familiar because it is quite similar to calculating area between two curves defined by rectangular equations. However, not all regions that we want to know the area of when dealing with two polar curves are that simple.

Sometimes we are not necessarily interested in an area between two polar curves, but an area shared by two polar curves (sometimes called the "common interior" for the curves). In these cases, a general formula or definite integral for area between two polar curves will not suffice. We cannot simply subtract the area under one curve from the area under the other. A little bit of intuition will be necessary to set up a proper definite integral (or sometimes multiple integrals) to calculate that area.

We discuss all of this and more in this new lesson of Calculus 2. Let's get to it!

In this lesson, you will learn:

• The general formula to calculate the area between two polar curves
• How to find points of intersection for two polar curves
• How to calculate the area of a common interior of two polar curves

I hope you find this video to be helpful!

-Josh