Math 1B
Overview: In this section we are going to find areas enclosed by polar curves. Here is a sketch of what the area that we will be finding in this section looks like:
The formula for finding this area is
𝐴 =
!
!
!
!
𝑓 𝜃 !
𝑑𝜃
This is often written as
𝐴 =
!
!
!
!
𝑟 !
𝑑𝜃 with the understanding that
Example: Find the area of the inner loop of 𝑟 = 1 + 2 cos 𝜃 . 𝑟 = 𝑓 ( 𝜃 ) .

To find the area of the region bounded by two polar curves, we use the following formula
𝐴 =
!
!
!
!
𝑔 𝜃 !
− 𝑓 𝜃 !
𝑑𝜃
Example: Find the area of the region that lies inside 𝑟 = 3 + 2 sin 𝜃 and outside 𝑟 = 2 .

Example: Set up the integral to find the area of the region that lies outside 𝑟 = 3 + 2 sin 𝜃 and inside 𝑟 = 2 .
Example: Find the area of the region enclosed by one loop of the curve 𝑟 = 2 sin 5 𝜃 .
Arc Length in Polar Coordinates: 𝐿 =
!
!
𝑟 !
+
!"
!"
!
𝑑𝜃
Example: Find the arc length of the polar curve 𝑟 = 𝜃 , 0 ≤ 𝜃 ≤ 1 .