Given an ellipse with a = semi-major axis and b = semi-minor axis, what is the exact formula for determining the average radius, i.e., the average distance between center and perimeter over the enter ellipse?
I see a few approximation formulas online. For example, the "arithmetic mean radius" is:
$$\frac{2a + b}{3}$$
And the "quadratic mean radius" is:
$$\sqrt{\frac{a^2 + b^2}{2}}$$
But these are just approximations. I want some formula, like an integral or a hypergeometric function, that gives an exact answer. It's easy to find one such formula for the circumference of an ellipse:
$$4a \int_0^{\pi/2} \sqrt{1 - (1 - b^2/a^2) \sin^2(x)} dx$$
Which given $e = \sqrt{1 - b^2/a^2}$, simplifies to:
$$4a \int_0^{\pi/2} \sqrt{1 - e^2 \sin^2(x)} dx$$
But I can't seem to find a working similar one for the mean radius. This answer seems to supply one, but it evaluates to exactly the semi-minor axis, which seems to be by definition not what I'm looking for.
EDIT: I specifically mean in terms of arc-length (thank you Parcly Taxel).