Area Under Polar Curves - Calculus 2

In Calculus 1 you learn that the area under a curve can be calculated using a definite integral. Up to this point, those curves that we have been calculating the area beneath were rectangular curves within the rectangular coordinate system, either represented by rectangular equations, or more recently in Calc 2, with parametric equations. However, we can also calculate the area under polar curves in the polar coordinate system.

When we calculated the area under rectangular curves, the definite integral represented the summation of the area of an infinite number of rectangles between the two values of x that we wanted to know the area of beneath the curve. This will be slightly different when calculating the area under polar curves, because the area under the curve will lie between two angles instead of two values of x. So, using an infinite number of rectangles no longer makes sense in the polar coordinate system. Instead, a different shape can be recognized to help us calculate the area.

Between two angles underneath a polar curve, we can add up the area of an infinite number of sectors to find the area beneath the curve. The area for a sector of a circle is equal to 1/2 times the radius squared times the angle of the sector. We can use this formula for area of a sector to help form the definite integral that will represent the area under a polar curve between two angles.

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 formula for calculating area under polar curves via integration
• How to determine the angles (bounds of integration) for area under polar curves
• How to use symmetry and solve definite integrals to find the area under polar curves

I hope you find this video to be helpful!

-Josh