# Activities

• • • ##### Subject Area

• Math: Calculus: Antiderivatives and Slope Fields

• ##### Author 9-12

60 Minutes

• ##### Software

TI InterActive!™

• ##### Other Materials
This is Activity 15 from the EXPLORATIONS Book:
TI InterActive! Math for High School

## Area Under the Curve

#### Activity Overview

In this activity, students explore the relationship between the area under the graph of a function and the integral of the function. They also discover the rule for the integral of f(x) = axn.

#### Before the Activity

• See the attached PDF file for detailed instructions for this activity
• Print pages 129-132 from the attached PDF file for your class
• #### During the Activity

Distribute the pages to the class.

• Define f(x) = x, 0 as the lower limit and 1 as the upper limit
• Enter different values of upper limit in L1 and compute the numerical integral of f(x) in L2
• Observe that the integral value, when the upper limit is 0, is 0 as area under the curve is 0
• Observe that when the upper limit is greater than the lower limit, the value of the numerical integral increases because the area is positive and increasing
• Observe that when the upper limit is greater than the lower limit, the value of the numerical integral increases because the area is getting larger, and the area is beneath the x-axis and summation is from right to left
• Define a function g(x) for the points plotted
• Define h(x) as the numerical integral of f(x), use FnInt function to find the numerical integral of f(x)
• Note, g(x) = h(x)
• Observe that the original function f(x) can be obtained by finding the derivative of the function g(x) or h(x)
• #### After the Activity

Students complete the Student Activity sheet.

Review student results:

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