graph the curve and find it's length, $r=\cos^2(\frac {\theta}{2}) $
I graphed it and found that it was a cardioid (or a sideways heart). I am getting stuck on the arc length.
this is what I have:
$$r=(\frac 12(1+\cos\theta) $$
$$\frac {dr}{d\theta}= -\frac 12\sin\theta $$
what I get for under the square root is :
$$\frac 14+\frac 12\cos\theta+\frac 14\cos^2\theta+ \frac 14\sin^2\theta $$
I ended up getting stuck with $\frac12+\frac12\cos\theta $
But I don't think this is right, where did I go wrong?
Best Answer
Things look fine. By the double-angle identity you have already used, the square root is $|\cos(\theta/2)|$. Integrate. Note the absolute value sign.