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)

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)   $r = a(1 + \sin \theta)$



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

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



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

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



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).

02 Area Bounded by the Lemniscate of Bernoulli r^2 = a^2 cos 2θ

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

Plane Areas in Polar Coordinates | Applications of Integration

The fundamental equation for finding the area enclosed by a curve whose equation is in polar coordinates is...

$\displaystyle A = \frac{1}{2}{\int_{\theta_1}^{\theta_2}} r^2 \, d\theta$


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