Question

Find the perimeter of the curve r=a(1-cos 𝜽)

Answer 1

Curve : r=a(1-cos 𝜽)

Perimeter of given curve is ,

`s=2xxint_0^pisqrt(r^2+((dr)/(d theta))^2d theta)`

`(dr)/(d theta)=a(sin theta)=>((dr)/(d theta))^2=a^2sin^2theta`

`r^2+((dr)/(d theta))^2=a^2[1-2cos theta+cos^2 theta]+a^2sin^2theta`

`sqrt(r^2+((dr)/(d theta))^2)=sqrt(2a)(1-cos theta)^(1/2)`

`=sqrt(2a)sqrt2sin(theta/2)`

`therefore s = 2int_0^pisqrt(2a)sqrt2sin(theta/2)d theta`

`=4aint_0^pisin(theta/2)d theta`

`=4a[-2cos(theta/2)]_0^pi`

∴ S = 8a