Question

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

Answer 1

Refer to explanation

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