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

1 Answer
Konstantinos Michailidis
Nov 7, 2015

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)]#

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