Circumference of the curve

Question

We have a cardioid curve. The length of the curve is given form the relation $\int_a^b |z'(t)|dt$.

How can we find the circumference of the cardioid curve???

Answer 1

$$\sqrt{(\cos 2t+\cos t)^2+(\sin t+2 \cos t \sin t)^2}\\=\sqrt{(\cos 2t+\cos t)^2+(\sin t+\sin{2t})^2}\\ =\sqrt{\cos^2{2t}+\cos^2{t}+2\cos{2t}\cos{t}+\sin^2{2t}+\sin^2{t}+2\sin{t}\sin{2t}}\\ =\sqrt{2+2(\cos{2t}\cos{t}+\sin{t}\sin{2t})}\\ =\sqrt{2+2\cos{(2t-t)}}\\=\sqrt{2+2(2\cos^2{(t/2)}-1)}=2|\cos{(t/2)}|$$

This can be integrated easily.

$$\int_0^{2 \pi} 2 |cos(t/2)|dt=\int_0^{\pi} 2 \cos{(t/2)}dt-\int_{\pi}^{2 \pi} 2 \cos{(t/2)}dt\\ =4\sin{(t/2)}|^{\pi}_0-4\sin{(t/2)}|^{2\pi}_{\pi}$$

Can you go from here?