How do you find the length of the cardioid #r=1+sin(theta)#?

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Answer 1 · 2015-09-13T15:48:52

Refer to explanation

The length of a curve with polar equation r = f(θ), a ≤ θ ≤ b, is given
by

#L=int_a^(b) sqrt(r^2+((dr)/(dθ))^2)dθ#

hence #r=1+sin(θ)# then

#(dr)/(dθ)=cos(θ)#

Then we have that

#L=int_a^(b) sqrt(r^2+((dr)/(dθ))^2)dθ=int_a^(b)(sqrt((1+sinθ)^2+cos^2θ)) dθ= int_a^(b) (sqrt2*sqrt(1+sinθ))dθ#

So if you have to calculated the last integral for known values of integration limits a,b