Education Technology

Riemann Sums and the Fundamental Theorem of Calculus

Published on 06/09/2008

Activity Overview

In this activity, students learn and explore how Indefinite integrals are used to find the antiderivative of a function. They also learn that Definite integrals can be used to find the area bounded by a function and the X-axis. From examples they learn that both types of integration can be connected together by the Fundamental Theorem of Integral Calculus.

Before the Activity

  • See the attached PDF file for detailed instructions for this activity
  • Print pages 39 - 48 from the attached PDF file for your class
  • During the Activity

    Distribute the pages to the class.

    Follow the Activity procedures:
    The Area Under a Parabola:
    Numerical Method:

  • Enter the function and graph it
  • Enter boundaries and number of rectangles
  • Use left-hand, right- hand, and midpoint Riemann Sums, and approximate the area bounded by the function


  • Analytical Method:
  • Define functions for the left-hand, right-hand, and midpoint rectangle methods
  • Evaluate the functions
  • Take the limit of each summation function as the number of rectangles approaches infinity
  • Find both approximate and exact values of the function for the area under the curve for the function
  • Notice that the area function is the antiderivative of f(x)
  • Find the area from a to b and predict the definite integral of the function with a and b as boundaries


  • Area Under other Curves:
  • Enter the function and find limits of the right-hand Riemann Sum to find the area from a to b
  • Compare the results with a corresponding definite integral
  • Notice that the limit of Riemann Sum is related to the antiderivative of the function
  • After the Activity

    Students will complete the practice exercise problems.

    Review student results:

  • As a class, discuss questions that appeared to be more challenging
  • Re-teach concepts as necessary