# Activities

• • • ##### Subject Area

• Math: Calculus: Applications of Integral, Antiderivatives and Definite Integrals

• ##### Author 9-12

60 Minutes

• ##### Device
• TI-89 / TI-89 Titanium
• ##### Accessories

CBL™/CBL 2™
Sensor - Temperature

• ##### Other Materials
This is Activity 6 from the EXPLORATIONS Book:
Advanced Placement Calculus with the TI-89.

The following materials are required for this activity:
• Ball
• Hot and cold water

## Applications of Integrals

#### Activity Overview

In this activity students investigate various applications of integration. They learn and explore to use analytic method as well as numerical, and graphical methods for solving various mathematical problems.

#### Before the Activity

• See the attached PDF file for detailed instructions for this activity
• Print pages 49 - 58 from the attached PDF file for your class
• For Newton's law of cooling activity, connect the Temperature Sensor into channel 1 of the CBL 2™ and start the DataMate program

#### During the Activity

Distribute the pages to the class.

Area between two Curves:

• Graph the two equations and find values for the left and right intersection points
• Evaluate the definite integral to find the area between the two curves

• Finding Arc Length:
• Use the built-in Arc feature to find the arc length
• Enter the equation and graph it
• Evaluate the expression for the definite integral and compare results

• First and second order differential equations:
• Toss a ball straight up from an initial height and at an initial velocity
• Find an equation to model the height of the ball over time

• Scatter Plots and Regression Curves:
• Use the Data/Matrix Editor and enter the data values
• Solve the equation and graph the data
• Find a quadratic regression equation and compare the equations

• Newton's Law of Cooling:
• Temperature probe is heated and then placed in cold water
• Measure the temperature after 30 seconds and find the temperature after 60 seconds
• Solve the equation and find the constant of proportionality

• Resistance Proportional to velocity:
• A ball is dropped from a height of 75 meters
• Find when the ball will hit the ground

• Logistic Growth:
• Given that the rate of growth of a population is directly proportional to both the population and the carrying capacity minus the population
• Determine the time period required to reach a certain population size by solving the differential equation
• #### 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