Chapter H.2, Problem 39E

To determine

To calculate:

The length of the following curve:

  $r=3\sin 2\theta$

Answer to Problem 39E

The length of the curve is $29.0714$

Explanation of Solution

Given information:

  $r=3\sin 2\theta$

Formula Used:

The length of the curve is given by the following expression:

  $L={\displaystyle \underset{0}{\overset{2\pi }{\int }}\sqrt{r^2+{\left(\frac{dr}{d\theta }\right)}^2}}d\theta$

Calculation:

The equation of the curve is:

  $r=3\sin 2\theta$

So, the curve is plotted as below:

  

The full length of the $r=3\sin 2\theta$ is in the interval of $0\le \theta \le 2\pi$

Know that:

  $\begin{array}{l}r=3\sin 2\theta \\ \frac{dr}{d\theta }=\frac{d}{d\theta }\left(3\sin 2\theta \right)\\ \frac{dr}{d\theta }=6\cos 2\theta \end{array}$

Now, the length of the given curve calculated as:

  $\begin{array}{l}L={\displaystyle \underset{0}{\overset{2\pi }{\int }}\sqrt{r^2+{\left(\frac{dr}{d\theta }\right)}^2}}d\theta \\ L={\displaystyle \underset{0}{\overset{2\pi }{\int }}\sqrt{{\left(3\sin 2\theta \right)}^2+{\left(6\cos 2\theta \right)}^2}}d\theta \\ L={\displaystyle \underset{0}{\overset{2\pi }{\int }}\sqrt{9{\sin }^{2}2\theta +36{\cos }^{2}2\theta }}d\theta \end{array}$

This is difficult to solve so, need to use the Simpson’s method for the function $f\left(\theta \right)=\sqrt{9{\sin }^{2}2\theta +36{\cos }^{2}2\theta }$

  $\begin{array}{l}n=10\\ \Delta \theta =\frac{2\pi }{10}=\frac{\pi }{5}\end{array}$

So, the subintervals are $\left[0,\frac{\pi }{5}\right]\left[\frac{\pi }{5},\frac{2\pi }{5}\right],\left[\frac{2\pi }{5},\frac{23\pi }{5}\right]\mathrm{...}\left[\frac{9\pi }{5},2\pi \right]$

Use the Simpson’s rule as:

  $\begin{array}{l}f\left(\theta \right)=\sqrt{80-64\cos 6\theta }\\ L=\frac{\Delta \theta }{3}\left[f\left(0\right)+4f\left(\frac{\pi }{5}\right)+2f\left(2\frac{\pi }{5}\right)+\mathrm{...}+4f\left(\frac{9\pi }{5}\right)+f\left(2\pi \right)\right]\\ L=\frac{\pi }{15}\left[f\left(0\right)+4f\left(\frac{\pi }{5}\right)+2f\left(2\frac{\pi }{5}\right)+\mathrm{...}+4f\left(\frac{9\pi }{5}\right)+f\left(2\pi \right)\right]\\ L=29.0714\end{array}$

Hence, the length of the curve is $L=29.0714$