Comparison of Sums
You'll probably get asked to compare the accuracy of different types of sums. Here are the main things you need to remember.
- Whether the left- and right-hand sums give over- or -underestimates depends on whether the function is increasing or decreasing.
- Whether the midpoint and trapezoid sums give over- or under-estimates depends on the concavity of the function.
- For any of these sums, the approximation gets more accurate as the number of sub-intervals gets bigger.
- The midpoint and trapezoid sums are more accurate than the left- and right-hand sums.
There are some things you need to know that we didn't include in the list. For example, if the function is increasing, is the LHS an over- or under-estimate? You should be able to figure that out
in about ten seconds by drawing this picture:
The rectangle doesn't cover enough area, so the LHS gives an underestimate. That means the RHS must be an overestimate.