Written by the Fiveable Content Team • Last updated September 2025
Written by the Fiveable Content Team • Last updated September 2025
Definition
An upper sum is an approximation of the area under a curve using the sum of areas of rectangles that lie above the curve over each subinterval. The height of each rectangle is determined by the maximum value of the function within that subinterval.
5 Must Know Facts For Your Next Test
Upper sums provide an overestimate of the area under a curve when approximating definite integrals.
The height of each rectangle in an upper sum is determined by the supremum (maximum) value of the function on that interval.
To calculate an upper sum, partition the interval into smaller subintervals and use the maximum function values within each subinterval to determine the heights.
Upper sums are part of Riemann sums, which also include lower sums, where lower sums use infimum (minimum) values for heights.
As the number of partitions increases (and their width decreases), the upper sum approaches the actual value of the definite integral if the function is integrable.
Review Questions
Related terms
Lower Sum: An approximation of area using rectangles below a curve, with heights determined by minimum function values in each subinterval.
Riemann Sum: A method for approximating integrals by summing up areas of multiple rectangles under a curve, including both upper and lower sums.
Definite Integral: The exact area under a curve within a given interval, calculated as a limit of Riemann sums as partition widths approach zero.