Problem:
Using both right-hand and left-hand endpoints, find the Riemann sums for the function

with n subintervals and simplify.


Solution:
  1. We first define Dx.

  2. We now define ai, i = 0, 1, 2, ..., n, the endpoints of these subintervals.

  3. We evaluate f(ai)

  4. We form the Riemann sum using left-hand endpoints:

  5. Simplifying, we obtain

  6. We form the Riemann sum using right-hand endpoints:

  7. Simplifying, we obtain