Chapter H.2, Problem 35E

To determine

To find: the exact length of the polar curve.

Answer to Problem 35E

Theexact length of the curve is $\pi$

Explanation of Solution

Given:

Given that the curve $r=3\sin \theta ,0\le \theta \le \pi /3$

Calculation:

The extra length of the curve $r=3\sin \theta$ is

  $L={\displaystyle \underset{0}{\overset{\frac{\pi }{3}}{\int }}\sqrt{r^2+{\left(\frac{dr}{d\theta }\right)}^2}}d\theta$

Now $r=3\sin \theta \Rightarrow \frac{dr}{d\theta }=3\cos \theta$

  $\begin{array}{c}L={\displaystyle \underset{0}{\overset{\frac{\pi }{3}}{\int }}\sqrt{9{\sin }^2\theta +{\left(3\cos \theta \right)}^2}d\theta }\\ ={\displaystyle \underset{0}{\overset{\frac{\pi }{3}}{\int }}\sqrt{9{\sin }^2\theta +3{\cos }^2\theta }d\theta }\\ ={\displaystyle \underset{0}{\overset{\frac{\pi }{3}}{\int }}\sqrt{9\left({\sin }^2\theta +{\cos }^2\theta \right)}d\theta }\\ ={\displaystyle \underset{0}{\overset{\frac{\pi }{3}}{\int }}\sqrt{9}d\theta }\left[\because {\sin }^2\theta +{\cos }^2\theta =1\right]\\ =\sqrt{9}{\left(\theta \right)}_0^{\frac{\pi }{3}}\\ =3\left(\frac{\pi }{3}-0\right)\\ =\pi \end{array}$

Conclusion:

Therefore, theexact length of the curve is $\pi$