Hi GeoMath!

The probability that "a" is greater than a randomly picked number is , which is the cumulative change to pick a number below "a".

The probability to pick a number between a and a+da is .

The probability to pick A>B is equal to the probability to pick A<B, so .

To calculate the expectation of A, you need:

So the expectation of A is:

To feed this to Wolfram|Alpha we apparently need that:

Then Wolfram|Alpha (link included) gives the result:

I believe this matches what you found empirically.