Question

Arc Lengths: Find the exact length of the polar curve r= 5sin(theta), 0<= theta <= pi/3.

Answer 1

The length L is given by \[ L = \int_{a}^{b} \sqrt{r^2 + (\frac{dr}{d\theta})^2}d\theta\]right?

Answer 2

So, r^2 = (5sin(theta))^2 and (dr/dtheta)^2 = (5cos(theta))^2

Answer 3

Now, the integral will be:\[\int\limits_{0}^{\pi/3} \sqrt{25(\sin^2{\theta} + \cos^2{\theta)}}d\theta = \int\limits_{0}^{\pi/3} \sqrt{25}d\theta\]Note that we want L > 0, so we pick the positive square root.

Answer 4

Wow, you are the best!!! Believe it or not, i literally JUST got that answer in an epiphany!!

Answer 5

Oh wait.....my answer is 5 right?? Wven though I have not done anything with the 0 and pi/3??

Answer 6

Nope, it should be: \(\LARGE \frac{5 \pi}{3} \) because you have to consider the limits of the integral (integral of dtheta = theta).

Answer 7

Haha, I forgot im still doing an integral. \[\int\limits_{?}^{?} \sqrt{25}=\int\limits_{?}^{?} 5x\]

Answer 8

and blah,blah. put your answer. Thank you sooo much!

Answer 9

No problem, mate. These arc length problems usually reduce to using some trig identity or trick.