Question 4.3.4: Computing Riemann Sums with Different Evaluation Points Comp...
Computing Riemann Sums with Different Evaluation Points
Compute Riemann sums for f(x)=\sqrt{x+1} on the interval [1, 3], for n = 10, 50, 100, 500, 1000 and 5000, using the left endpoint, right endpoint and midpoint of each subinterval as the evaluation points.
The numbers given in the following table are from a program written for a programmable calculator. We suggest that you test your own program or one built into your CAS against these values (rounded off to six digits).
| n | Left Endpoint | Midpoint | Right Endpoint |
| 10 | 3.38879 | 3.44789 | 3.50595 |
| 50 | 3.43599 | 3.44772 | 3.45942 |
| 100 | 3.44185 | 3.44772 | 3.45357 |
| 500 | 3.44654 | 3.44772 | 3.44889 |
| 1000 | 3.44713 | 3.44772 | 3.44830 |
| 5000 | 3.44760 | 3.44772 | 3.44783 |
There are several conclusions to be drawn from these numbers. First, there is good evidence that all three sets of numbers are converging to a common limit of approximately 3.4477. Second, even though the limits are the same, the different rules approach the limit at different rates. You should try computing left and right endpoint sums for larger values of n, to see that these eventually approach 3.44772, also.