integration of polar area

Problem
Find the area bounded by $r = a \sin 3\theta$ and $r = a \cos 3\theta$.
 

Problem
Find the area bounded by $r = a \sin 2\theta$ and $r = a \cos 2\theta$.
 

Example 6
What is the area within the curve r² = 16 cos θ?
 

Example 4
Find the area of the inner loop of the limacon r = a(1 + 2 cos θ).
 

Example 3
Find the area inside the cardioid r = a(1 + cos θ) but outside the circle r = a.
 

Problem
Find the area individually enclosed by the following Cardioids:
(A)   $r = a(1 - \cos \theta)$
(B)   $r = a(1 + \cos \theta)$
(C)   $r = a(1 - \sin \theta)$
(D)   $r = a(1 + \sin \theta)$
 

Example 2
Find the area bounded by the lemniscate of Bernoulli r² = a² cos 2θ.
 

Plane Areas in Rectangular Coordinates

There are two methods for finding the area bounded by curves in rectangular coordinates. These are...

1. by using a horizontal element (called strip) of area, and
2. by using a vertical strip of area.

The strip is in the form of a rectangle with area equal to length × width, with width equal to the differential element. To find the total area enclosed by specified curves, it is necessary to sum up a series of rectangles defined by the strip.