Find the exact length of the curve. Use a graph to determine the parameter interval.

r=cos^2(theta/2)

Question

Find the exact length of the curve. Use a graph to determine the parameter interval.  

r=cos^2(theta/2)

anonymous · Answer

The formula for the length of a polar curve is $$\int\limits_{a}^{b} \sqrt{r^2+(dr/d\theta)^2}$$

anonymous · Answer

Do you see how to write the integrand?

anonymous · Answer

Heres what I got using that, which gave me a wrong answer...

$$\int\limits_{0}^{4\pi}\sqrt{(\cos ^{4}(\theta/2)+(\sin^{2}(\theta))/4}$$

anonymous · Answer

I think the second term should just be sin^2(theta)

anonymous · Answer

$$r=\cos^{2}(\theta/2), r^{2}=\cos^{4}(\theta/2)$$

anonymous · Answer

No wait, you were right about that derivative

anonymous · Answer

okay, should i use half angle formula for the r^2 piece under the sqrt?

anonymous · Answer

You shouldn't need to, you're just evaluating this numerically, right?

anonymous · Answer

yea, maybe my bounds are incorrect, i used wolfram to get the bounds

anonymous · Answer

Wolfram tends to be correct, and 4pi does make sense.

anonymous · Answer

so why is my answer of 8 incorrect?

anonymous · Answer

Actually, wolfram says it should be 0 to 2pi for me

anonymous · Answer

yep that was the problem, thanks for the help

anonymous · Answer

No problem

anonymous · Answer

How did you guys do the integral?