I’ve been following along with the book ”Differential geometry of curves and surfaces” and I came along a question asking to solve for the distance travelled of a point by a rolling circle within one full rotation.
I solved for the arclength integral and got an indefinite integral which looked something like
$-4r\cos(t/2)+C$ where t is the time
Obviously if we evaluate this from $0$ to $2\pi$ we would get 8r. However my confusion comes from the fact that if we evaluate this from $0$ to $4\pi$ instead of getting 16r we get 0
Shouldn’t the indefinite integral of the expression be a function giving the arclength over some interval? Arclength can’t be negative so how come the function representing the arclength is periodic rather than tending to infinity? Why would we get 0 rather than 16r for the amount of distance travelled over two periods?