Education Technology

Calculus: Riemann Sums
by Texas Instruments

Updated on October 02, 2015


  • Learn that for a continuous nonnegative function f, there is one interpretation of the definite integral f(x)dx from a to b, the area of the region R, bounded above by the graph of y = f(x), below by the x-axis, and by the lines x = –a and x = –b
  • Visualize and compute values for three different Riemann sums: left-hand endpoint, right-hand endpoint, and midpoint, and use these values to estimate the area of a region R
  • Learn about the nature of these estimates as the number of rectangles increases
  • Consider other functions and relate these Riemann sums to function characteristics


  • Riemann sum
  • Left-hand endpoint, right-hand endpoint, and midpoint sum
  • Area of a plane region
  • Underestimate, overestimate
  • Definite integral
  • Continuous

About the Lesson

This lesson involves three Riemann sums used to estimate the area of a plane region. As a result, students will:

  • Conjecture about each estimate as the number of rectangles increases.
  • Conjecture about each estimate in relation to certain characteristics of the function.
  • Consider the magnitude of the error in each approximation.
  • Conjecture about other geometric figures that might produce better estimates.