Upper sum

Definition

The upper sum is an approximation of the area under a curve using the sum of the areas of rectangles that overestimate the region. Each rectangle's height is determined by the maximum function value over a subinterval.

5 Must Know Facts For Your Next Test

1. Upper sums are used to approximate definite integrals from above.
2. The interval $[a, b]$ is divided into $n$ subintervals for calculating upper sums.
3. The height of each rectangle in the upper sum corresponds to the supremum (maximum) value of the function within that subinterval.
4. As $n$ approaches infinity, the upper sum approaches the exact value of the integral if the function is integrable.
5. Upper sums are often compared with lower sums to establish bounds for Riemann sums.

Review Questions

• What does each rectangle's height represent in an upper sum?
• How does increasing the number of subintervals affect the accuracy of an upper sum?
• In what scenario would you use an upper sum over other types of Riemann sums?

Related terms

Lower Sum:

An approximation method where each rectangle's height is determined by the minimum function value over a subinterval, underestimating the area.

Riemann Sum:

A method for approximating definite integrals by summing up areas of rectangles based on function values at specified points within subintervals.

Definite Integral:

The exact area under a curve between two points on a graph, typically calculated using limits and Riemann sums.