Can someone please explain to me how express the length of a curve in terms of the elliptic integral:
$$E(k) = \int_0^{\pi/2} \sqrt{1-k^2\sin^2t} \, dt$$
for $0 < k < 1$.
For example, how would one express an ellipse with semi-major axis $a$ and semi-minor axis $b$?
This was given to me as a question on a problem set for an Analysis Course but I've never heard of the elliptic integral and furthermore, it isn't mentioned anywhere in my textbook (Munkres, Analysis on Manifolds) and hasn't been mentioned in class.
Can someone please help to explain this?
Thank you kindly.