Question

How do you find the exact length of the polar curve `r=3sin(theta)` on the interval `0<=theta<=pi/3` ?

Answer 1

The arc length is `pi`.

Let us look at some details.

`r=3sin theta`

by differentiating with respect to `theta`,

`Rightarrow {dr}/{d theta}=3cos theta`

So, the arc length L can be found by

`L=int_0^{pi/3}sqrt{r^2+({dr}/{d theta})^2}d theta`

`=int_0^{pi/3}sqrt{3^2sin^2theta+3^2cos^2theta}d theta`

by pulling `3` out of the square-root,

`=3int_0^{pi/3}sqrt{sin^2theta+cos^2theta}d theta`

by `sin^2theta+cos^2theta=1`,

`=3int_0^{pi/3}d theta=3[theta]_0^{pi/3}=3(pi/3-0)=pi`