The result seems ok (to me, that is). But you have done more work than you were asked to do. In other cases, it may be a terrible amount of work to evaluate the limit of the Riemann sums directly, instead of evaluating the corresponding integral.
Assuming all of that is correct, the lim as n approaches infinity is 1/2. However, I've spent a lot of time looking at this problem and might be completely off track.