# Arc length of polar curves

##### Intros

##### Examples

###### Lessons

**Finding the Arc Length of Polar Equations**

Find the length of the curve $r=4 \sin \theta$ from $0 \leq \theta \leq \pi$.- Find the length of the curve $r=e^{\theta}$ from $0 \leq \theta \leq 3$.
- Find the length of the curve $r=\theta^2$ from $0 \leq \theta \leq 1$.
- Find the length of the curve $r=3^{\theta}$ from $0 \leq \theta \leq \pi$. (Hint: $\int_{a}^{b} C^xdx=\frac{c^x}{\ln(c)}|_{a}^{b}$ where C is a constant)

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###### Topic Notes

In this lesson, we will learn how to find the arc length of polar curves with a given region. We will first examine the formula and see how the formula works graphically. Then we will apply the formula to some of the questions below. Make sure you know trigonometric identities very well, as you will often need to use substitution to make your integrals simpler to integrate.

Let $r=f(\theta)$
be a polar curve and $\alpha \leq \theta \leq \beta$.

Then we use the following formula to calculate the arc length of the curve:

$L=\int_{\alpha}^{\beta}\sqrt{r^2+(\frac{dr}{d\theta})^2}d\theta$

Then we use the following formula to calculate the arc length of the curve:

$L=\int_{\alpha}^{\beta}\sqrt{r^2+(\frac{dr}{d\theta})^2}d\theta$

###### Basic Concepts

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