# How do I find the length of the cardioid r=1+sintheta?

The length of the cardiod $r = 1 + \sin \theta$ between a and b is
$L = {\int}_{a}^{b} \left(\sqrt{{r}^{2} + {\left(\frac{\mathrm{dr}}{d \left(\theta\right)}\right)}^{2}}\right) \left(d \left(\theta\right)\right)$
hence $\frac{\mathrm{dr}}{d \left(\theta\right)} = \cos \theta$ and
L=int_a^b (sqrt(1+sin^2theta+2costheta+cos^2theta))d(theta)=> L=sqrt2*int_a^b sqrt(1+costheta)d(theta)=> L=sqrt2[(2sqrt(1+costheta))*tan(theta/2)]_a^b=> L=2sqrt2[sqrt(1+cosb)*cos(b/2)-sqrt(1+cosa)*cos(a/2)]