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Activity Overview

In this activity, students create a general solution for calculating the ratio of the areas above and below a curve.

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Before the Activity

Download TI Connect™ using the TI Connectivity Cable
See the attached PDF file for detailed instructions for this activity
Print pages 1 - 4 from the attached PDF file for your class
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During the Activity

Distribute the pages to the class.

Follow the Activity procedures:

Plot the lines y = x, y = 1, and x = 1
Form a square using segments on the lines y = 1 and x = 1
Evaluate the ratio of the area of the triangle above the y = x line to the area of the triangle below the y = x line
Notice that the ratio is 1 : 1
Plot the curves y = x^{2}, y = 1, and x = 1
Evaluate the areas above and below the curve by integrating them numerically
Note that the ratio of the areas is 2 : 1
Understand that if the index of x in the curve is n, then the ratio of the areas of the curves will be n : 1
Define some functions and verify the conjecture
Realize that if the index of x is a negative integer, there is no area enclosed under the graph in the specified interval
Repeat the problem with different values of n over various intervals
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After the Activity

Students answer the questions on the activity sheet.

Review student results:

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