# Find the exact length of the parametric curve(Not sure what I'm doing wrong)

As the title says, I'm not sure what I'm doing wrong. Any help would be greatly appreciated. Here's the problem with my solution.

Find the exact length of the parametric curve $(x,y)=(\theta+\sin \theta,−\cos θ)$, where $0\le θ \le \pi$.

Solution: $$\frac{dx}{d\theta}=1+\cos \theta$$ $$\frac{dy}{d\theta}=\sin \theta$$

Using the formula $$\int_0^\pi \sqrt{\left( \frac{dx}{d\theta} \right)^2+ \left( \frac{dy}{d\theta} \right)^2} \, d\theta$$

I have $$\int_0^\pi \sqrt{2+2\cos \theta} \, d\theta$$

and then $$\left[ \frac{(4+4\cos \theta)\sqrt{2+2\cos \theta}}{3} \right]_{0}^{\pi}$$

I end up with $\dfrac{16}{3}$!

• I'm not sure how you integrated, but try using the substitution $t=\tan\frac{\theta}{2}.$ Also, make what you're integrating with respect to more explicit please. Mar 8, 2017 at 4:35
• What I did was (2+2cos(theta))^(3/2))/(3/2) as my integration technique. Mar 8, 2017 at 4:41
• @Infremo Maybe check Wolfram alpha to see if your in-definite integral is right. (It is not. That's not a valid integration rule.) Mar 8, 2017 at 4:55
• @ Infremo It is a cycloid, obtainable by $x$ shift and $y$ reflection.. standard references would be of help, $\theta = 2 \phi$ substitution is also helpful Mar 8, 2017 at 10:09
• @Infremo Yeah, you can't integrate (crud)$^n$ as (crud)$^{n+1}/(n+1)$, unless (crud)$=x$. There's some chain-rule-y stuff you have to unwind. Mar 8, 2017 at 14:09

Hint

First recall that integration is harder than differentiation and that you can't just use the power rule when there's a cosine running around inside (remember the chain rule? It's not as easy in reverse.)

As for how to do the integral, there's a popular trig identity involving $\sqrt{1+\cos(x)}$

Hint: $$2+2\cos\theta=2(1+\cos\theta)=2(2\cos^2\dfrac{\theta}{2})=4\cos^2\dfrac{\theta}{2}$$