Question

Question #d0a78

Answer 1

`L = int sqrt(dx^2 + dy^2)`

Explanation:

`L = int sqrt((dx/(d theta))^2 + (dy/(d theta))^2 )d theta`
`L = int sqrt((d(r cos theta) /(d theta))^2 + (d(r sin theta) /(d theta))^2 )d theta`
`L= int sqrt(((dr)/(d theta))^2 + r^2) d theta`
now
`r = 2(1+cos(theta))`
so
`(dr)/(d theta) = -2 sin(theta)`
`r^2 = 4(1 + 2 cos theta + cos^2 theta)`
Adding yields
`4 sin^2 theta + 4 cos^2 theta + 4 + 8 cos theta = 8(1+ cos theta)= 16 cos^2(theta/2)`
so
`L=int 4 cos(theta/2) d theta`
`L=8 sin(theta/2)`