TITLE: The average radius of an ellipse
QUESTION [7 upvotes]: I have recently been studying the 3rd Kepler´s law which uses the average radius of an orbit. I would love to know how to find the geometric average radius of an ellipse. Would you please show me or at least help me with the process of integration?

REPLY [2 votes]: @Martin-Blas has already suggested two possible notions of average

Take an average with respect to $\theta$, the central angle
Take an average with respect to time, i.e.,$$
\frac{1}{T}\int_0^T \| u(t) \| dt,
$$
where $T$ is the period, and $u(t)$ is the position of the planet at time $t$, and the sun is located at the origin of the coordinate system. 

There's a third notion, which is 

An average with respect to arclength, i.e., $$
\int_0^L \| u(t) \| ds$$
where $L$ is the total arclength of the ellipse, and $ds = \|u'(t)\| dt$ is  arclength integrand. 

I'm pretty certain that one could even hoke up some others, but the key thing here is the one mentioned by others: this question doesn't have an answer until you know the measure with respect to which the "average" is being computed.