Question

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

Answer 1

The length of the cardiod `r=1+sintheta` between a and b is

`L= int_a^b (sqrt(r^2+((dr)/(d(theta)))^2))(d(theta))`

hence `dr/(d(theta))=costheta` 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)]`