6.1: The Area Between
This activity is intended to supplement Calculus, Chapter 5, Lesson 1.
ID: 9983
Time required: 45 minutes
Topic: Applications of Integration
- Calculate the area enclosed by two intersecting curves defined in Cartesian coordinates.
- Use the Solve and Integral commands to verify the manual computation of areas bounded by curves.
Activity Overview
In this activity, students will use the TI-89 graphing calculator to find the area between two curves while determining the required amount of concrete needed for a winding pathway.
Teacher Preparation
The students should already be familiar with the concept of the integral as well as using them to find the area below a curve.
- Students should know how to use the Integral and solve commands.
- The screenshots on page 1 demonstrate expected student results.
Classroom Management
- This activity is designed to be student-centered. You may use the following pages to present the material to the class and encourage discussion. Students will follow along using their calculators.
- The students will need to be able to compute the integrals and solve equations using their calculator on their own. You may choose to have students compute these by hand.
- Before starting this activity, students should go to the home screen and select \begin{align*}F6\end{align*} :Clean Up > 2:NewProb, then press \begin{align*}\div\end{align*}. This will clear any stored variables, turn off any functions and plots, and clear the drawing and home screens.
Associated Materials
- Student Worksheet: The Area between Two Curves http://www.ck12.org/flexr/chapter/9730
Problem 1 – Making Pathways
The first problem involves a pathway with sine functions as the borders. Students are to graph the functions in \begin{align*}y1\end{align*} and \begin{align*}y2\end{align*}. Then they can use the Integral tool to calculate the area under each curve. Students will be asked to enter the lower and upper limits. They can enter \begin{align*}-2 \pi\end{align*} and \begin{align*}2 \pi\end{align*} directly, instead of making a guess with the arrow keys.
Students are to go to the Home screen, where they will calculate the area between the curves using the nInt command. Then they need to multiply their answer by \begin{align*}\frac{1}{3}\end{align*} to find the volume of the pathway or amount of concrete needed \begin{align*}(12.5664 \ ft^3)\end{align*}.
Problem 2 – Finding New Pathways
The second problem involves a pathway with cubic functions as the borders. Students are to find the volume of a new pathway using the graph and the calculator. They should use the same method in the previous problem.
Remind students that when finding the area of the region, they must always take the integral of the top function minus the bottom one (this is where many mistakes happen).
Problem 3 – Stepping Stones
In this problem, students will find the volume of one stepping stone. Students are to graph the functions and then find the intersection points using the Intersection tool. These intersection points will be the lower and upper limits. Students can also use the Solve command on the Home screen; however it will give three values. They need to use the two positive values.
Then, students can find the volume of the stepping stone using the previous method. They may think that \begin{align*}g(x)\end{align*} is the top function, but \begin{align*}f(x)\end{align*} is the top function in the interval.
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