# Find the arc length of a polar function from 0 to 2pi

## Homework Statement

This is another problem my teacher game me.

Given the Polar function r=6*sin(t/2) where the variable t is the angle theta in radians, and that t is between 0 and 2*Pi inclusive. Find the distance around the perimeter of the graph. Hint: this is arc length , round to the nearest integer.

## Homework Equations

I know to do this to calculate the arc length... it think integrate the equation from 0 to 2$$\pi$$

## The Attempt at a Solution

integrate(6 sin(t/2) dt, 0, 2pi)
= -12 cos(t/2) ] 0, 2pi
= -12 cos(2pi/2) - (-12 cos(0/2))
= -12 cos(pi) - (-12 cos(0))
= -12 (-1) - (-12 (1)) = 24

I also used f-int in my calculator and got 24. Am I missing a step, what am I doing wrong?

Char. Limit
Gold Member
You know the formula for the arclength of a polar function, right?

$$S = \int_a^b \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2} d\theta$$

Tell me if I'm mistaken but that's... the equation squared plus its derivative squared... right sure making sure.

Char. Limit
Gold Member
You are not mistaken. That really is the correct formula for the arc-length of a polar function.