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