I suspect apmcavoy was thinking of "elliptic functions", as in number theory, which are quite different. In any case, he is wrong. Elliptic integrals did, indeed, arise from trying to find the arc length of an ellipse which is NOT as simple as he seems to think. The arclength of an ellipse cannot be written in any simple formula.
(The
area is very simple. The area of the ellipse

is just

. The distance around (circumference?) an ellipse is not.)