Arc length of polar curves

Arc length of polar curves

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.

Lessons

Let r=f(θ)r=f(\theta) be a polar curve and αθβ\alpha \leq \theta \leq \beta.
Then we use the following formula to calculate the arc length of the curve:
L=αβr2+(drdθ)2dθL=\int_{\alpha}^{\beta}\sqrt{r^2+(\frac{dr}{d\theta})^2}d\theta
  • Introduction
    Arc Length of Polar Curves Overview:

  • 1.
    Finding the Arc Length of Polar Equations
    Find the length of the curve r=4sinθr=4 \sin \theta from 0θπ0 \leq \theta \leq \pi.

  • 2.
    Find the length of the curve r=eθr=e^{\theta} from 0θ30 \leq \theta \leq 3.

  • 3.
    Find the length of the curve r=θ2r=\theta^2 from 0θ10 \leq \theta \leq 1.

  • 4.
    Find the length of the curve r=3θr=3^{\theta} from 0θπ0 \leq \theta \leq \pi. (Hint: abCxdx=cxln(c)ab\int_{a}^{b} C^xdx=\frac{c^x}{\ln(c)}|_{a}^{b} where C is a constant)