Education Technology

Exploring the Fundamental Theorem of Calculus

Published on 06/09/2008

Activity Overview

In this Derive™ activity, students investigate the Fundamental Theorem of Calculus and explore examples of Riemann Sums for approximating the Definite Integral: the Midpoint Sum, the Left Hand Endpoint Sum, the Right Hand Endpoint Sum, The Trapezoidal Sum, and Simpson's Approximating Sum.

Before the Activity

  • See the attached DFW file for detailed instructions for this activity
  • Print pages from the attached DFW file for your class
  • During the Activity

    Distribute the pages to the class.

    Follow the Activity Procedures:

  • Set up a 2D plot window
  • Construct the expressions for the Midpoint Sum and other Riemann Sums, in stages


  • Plot a polynomial function
  • Observe an expression whose graph is a sequence of rectangles over an interval having the value of the function at the midpoint of each interval as the height of the rectangle above that interval
  • Graph the result
  • Evaluate the area lying between the resulting graph and the x-axis
  • Realize that if a function is positive over an interval, the Midpoint Sum is the area of a sequence of rectangles (constructed as illustrated in the graph)
  • Notice that the area of the rectangles is almost the same as the area between the graph and the x-axis
  • Understand that the definite integral of a function over an interval is the difference of the area of the region lying above the x-axis and below the graph of the function, and the area of the region lying below the x-axis and above the graph of the function


  • Use the Derive software to compute the value of an integral over an interval, and compare the result with the value calculated with the help of the Midpoint Sum formula


  • Graph a polynomial function
  • Set the lower value of the interval as a constant and let the upper value be a variable
  • Use the formula for the Midpoint Sum to approximate the integral
  • Record the action of the Midpoint Sum when the value of the function is zero, negative, positive, minimum, and maximum


  • Understand that as the limit of the maximum width of the subintervals making up the partition goes to zero, the variable sum will have the function as its derivative
  • After the Activity

    Students answer questions listed on the activity sheet.

    Review student results:

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