Question

the spiraling polar curve

r=5e^20
from 0 to 2pi

how do you find the length?

Answer 1

If you call a length element on the curve ds, then we have:

(ds)^2 = (dr)^2 + r^2 (dtheta)^2 (1)

dr is the component of the length element in the radial direction and r d theta is the component in the direction orthogonal to that. So, by Pythagoras' therem you have to ad up the squares of both to find the squared length element.

From (1) it follws that:

ds = sqrt[(dr)^2 + r^2 (dtheta)^2] =

sqrt[r^2 + (dr/dtheta)^2] dtheta


If you integrate this from theta = 0 to 2 pi, you get the curve length:

s = Integral from zero to 2 pi of

sqrt[r^2 + (dr/dtheta)^2] dtheta