# Activities

• • • ##### Subject Area

• Math: Calculus: Antiderivatives and Slope Fields

• ##### Author 9-12

60 Minutes

• ##### Device
• TI-89 / TI-89 Titanium
• ##### Other Materials
This is Activity 5 from the EXPLORATIONS Book:
Advanced Placement Calculus with the TI-89.
• ##### Report an Issue

Riemann Sums and the Fundamental Theorem of Calculus

#### 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