# Polar Curves

## Length of Arc in Polar Plane | Applications of Integration

The length of arc on 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. See it here: http://www.mathalino.com/reviewer/integral-calculus/length-arc-polar-pla...

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

**Example 6**

What is the area within the curve r^{2} = 16 cos θ?

## 05 Area Enclosed by Four-Leaved Rose r = a cos 2θ

Find the area enclosed by four-leaved rose r = a cos 2θ.

## 04 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

**Example 3**

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

## 01 Area Enclosed by r = 2a sin^2 θ

**Example 1**

Find the area enclosed by *r* = 2*a* sin^{2} 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...

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