- 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

Midpoint sum notation isn't as concise as left-hand sums and right-hand sum notation. However, we can still use similar shortcuts to find the values of *f* at all the points we need, add these values up, and then multiply by the width of a sub-interval. Check out our examples!

Example 1

Let |

Exercise 1

Use a midpoint sum with 5 sub-intervals to estimate the area between *f* (*x*) =* x*^{2} and the *x*-axis on [0,1].

Exercise 2

Use a midpoint sum with 10 sub-intervals to estimate the area between *f* (*x*) = *x*^{2} and the *x*-axis on [0,1].