What's a decent formula for the average distance of an orbiting body, as a function of periapsis, apoapsis?

How do you calculate the average distance (over TIME) of an orbiting body from its primary, given the periapsis and apoapsis (or equivalently, the orbit's semimajor axis and its eccentricity)?

I know in abstract that the solution should be:

R_avg = ∫R(t)dt / T

where R(t) is the body's distance as a function of time; and T is its period. However, I don't think there's a closed-form solution for R(t), is there? Failing that, is there a (short) formula that gives a fair numerical approximation?

Also, my intuition is that "T" should cancel out of the final equation (i.e. R_avg should depend just on the geometry of the orbit), since the geometry uniquely determines the fraction of time that the body spends in any particular segment of the orbit.