Length of a Curve in Polar Coordinates Example
\[r=R( \theta)\]
the length of the curve between the points \[(r_1, \theta_1)\]
and \[r_2, \theta_2\]
is \[L= \int^{ \theta_2}_{\theta_1} \sqrt{r^2+ (\frac{dr}{d \theta})^2}d \theta\]
.Example: If
\[r= cos \theta , \; - \frac{\pi}{2} \le \theta \lt \frac{\pi}{2}\]
find the length of curve.Differentiate
\[r=cos \theta \rightarrow \frac{dr}{d \theta} = - sin \theta\]
.\[\begin{equation} \begin{aligned} L &= \int^{\frac{\pi}{2}}_{- \frac{\pi}{2}} \sqrt{cos^2 \theta + (-sin \theta)^2} d \theta \\ &= \int^{\frac{\pi}{2}}_{- \frac{\pi}{2}} 1 d \theta \\ &= [ \theta ]^{\frac{\pi}{2}}_{- \frac{\pi}{2}} \\ &= (\frac{\pi}{2})- (- \frac{pi}{2}) = \pi \end{aligned} \end{equation}\]
