Find arc length of cardioid $\rho=a(1+\text{cos}\psi)$

Question

Find the arc length of the cardioid $\rho=a(1+\text{cos}\psi)$ where $a>0,\:\:0\le\psi\le2\pi$

My work:

We calculate that $\rho_{\psi}'=-a\:\:\text{sin}\psi$ $\implies$ $$\sqrt{(\rho_\psi')^{2}+\rho^2}=$$ $$\sqrt{2a^2(1+\text{cos}\psi)}=$$ $$\sqrt{4a^2\:\: \text{cos}^2\left(\frac{\psi}{2}\right)}=$$ $$2a|\text{cos}\left(\frac{\psi}{2}\right)|$$

I'm stuck after this. Any help is greatly appreciated.

Answer 1

Continue with $$L=\int_0^{2\pi} 2a|\cos\frac\psi2|d\psi=4a \int_0^{\pi} \cos\frac\psi2d\psi=8a $$