Integration of Polar Area

03 Area Enclosed by Cardioids: r = a(1 + sin θ); r = a(1 - sin θ), r = a(1 + cos θ), r = a(1 - cos θ)

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)   $r = a(1 + \sin \theta)$
 

003-cardioid-neg-pos-sine-cosine.gif

 

08 Area Enclosed by r = a sin 3θ and r = a cos 3θ

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

008-polar-area-three-leaf_rose_sine_cosine.gif

 

05 Area Enclosed by r = a sin 2θ and r = a cos 2θ

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

008-polar-area-four-leaf_sine_cosine.gif

 

06 Area Within the Curve r^2 = 16 cos θ

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

04 Area of the Inner Loop of the Limacon r = a(1 + 2 cos θ)

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

03 Area Inside the Cardioid r = a(1 + cos θ) but Outside the Circle r = a