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

Let us look at some details.

So, the arc length L can be found by

To find the exact length of the polar curve (r = 3\sin(\theta)) on the interval (0 \leq \theta \leq \frac{\pi}{3}), you can use the arc length formula for polar curves:

[ L = \int_{\alpha}^{\beta} \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2} , d\theta ]

Where:

• ( \alpha ) and ( \beta ) are the starting and ending angles of the interval,
• ( r ) is the polar function,
• ( \frac{dr}{d\theta} ) is the derivative of the polar function with respect to ( \theta ).

First, find ( \frac{dr}{d\theta} ): [ \frac{dr}{d\theta} = 3\cos(\theta) ]

Now, plug the values into the arc length formula and integrate over the given interval (0 \leq \theta \leq \frac{\pi}{3}):

[ L = \int_{0}^{\frac{\pi}{3}} \sqrt{(3\sin(\theta))^2 + (3\cos(\theta))^2} , d\theta ]

[ L = \int_{0}^{\frac{\pi}{3}} \sqrt{9(\sin^2(\theta) + \cos^2(\theta))} , d\theta ]

[ L = \int_{0}^{\frac{\pi}{3}} \sqrt{9} , d\theta ]

[ L = 3\int_{0}^{\frac{\pi}{3}} , d\theta ]

[ L = 3\left[\theta\right]_{0}^{\frac{\pi}{3}} ]

[ L = 3\left(\frac{\pi}{3} - 0\right) ]

[ L = \pi ]

So, the exact length of the polar curve ( r = 3\sin(\theta) ) on the interval ( 0 \leq \theta \leq \frac{\pi}{3} ) is ( \pi ).