- Topics At a Glance
- Left-Hand Sum
- Right-Hand Sum
- Comparing Right- and Left-Hand Sums
- Error in Left- and Right-Hand Sums
- Midpoint Sum
- Midpoint Sums with Shortcuts
- Over or Under Estimates
- Trapezoid Sum
- Trapezoid Sum with Shortcuts
- Over or Under Estimates
**Comparison of Sums**- Definite Integrals of Non-Negative Functions
- Definite Integrals of Real-Valued Functions
- Conditions for Integration
- General Riemann Sums
- Properties of Definite Integrals
- Single-Function Properties
- Talking About Two Functions
- Thinking Backwards
- Average Value
- Averages with Numbers
- Averages with Functions
- In the Real World
- I Like Abstract Stuff; Why Should I Care?
- How to Solve a Math Problem

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.