Polar Curves

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)   $r = a(1 + \sin \theta)$



01 Area Enclosed by r = 2a cos^2 θ

Find the area enclosed by r = 2a cos2 θ.



Length of Arc in Polar Plane | Applications of Integration

The length of arc in polar plane is given by the formula:

$\displaystyle s = \int_{\theta_1}^{\theta_2} \sqrt{r^2 + \left( \dfrac{dr}{d\theta} \right)^2} ~ d\theta$




The formula above is derived in two ways.

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).

01 Area Enclosed by r = 2a sin^2 θ

Example 1
Find the area enclosed by r = 2a sin2 θ.

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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