Area of a region bounded by a polar curve (continued)

 In the previous post, we set  the formula to find the area of a region bounded by a polar curve. Let's do an application.

Example. Find the area of one petal of the rose defined by the equation r = 3sin (2θ)

Solution

Here is the graph of the of the petal of the rose

The first petal of the rose is traced out from the polar coordinates (0, 0) and (0, 𝝅/2). To find the area inside the petal, let's use the formula of the area of the region bounded by a polar curve. In this formula we substitute 𝛼 by 0 and 𝛽 by 𝝅/2. 

To evaluate this integral, let's use the formula sin²𝛂 = 1 - cos(2𝛂) with 𝛂 = 2𝜃

Practice

Find the area inside the cardioid defined by the equation r = 1-cos𝜃