TEMPLATE LINK FOR TEACHERS TO CONTRIBUTE:
SolidWorks Lesson Template for Teachers to Contribute
Cover Sheet for Exemplary Lessons/Units Project
Faculty Member Name: M. Planchard Date: 5/06/06
Organization: SolidWorks Corporation
Title of Lesson/Unit: Parabola – From Algebra to Calculus
Applicable for: All math and science classes requiring imported data
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Science, Technology, Engineering and Math) STEM Concepts Addressed: This lesson develops two examples with a parabola. The first example explores graphing a parabola from an equation, finding the area bounded by the x axis, center of mass, and volume of a solid in revolution about a line parallel to the x-axis. Understanding a parabola in algebra can lead to success in Calculus and Calculus based science classes such as physics, engineering mechanics and dynamics. The second examples bounds the parabola by both the x and y axis, explores the centroid and volume in revolution.
Length of instruction period: 50 minutes
How many periods needed to implement lesson unit: 1
Grade Level(s) for use: all
Objectives:
1. Understand the relationship between different coefficients of a parabola.
2. Calculate area under the curve.
3. Calculate the center of mass and understand its definition.
4. Able to revolve a closed area to create a solid and calculate its volume.
For a parabola, there should first be a physical sense to where the center of mass is located on a body – before all the theoretical calculations. Is there symmetry? If the answer is yes than one of the components of the center of mass must be on an axis.
The equation illustrated is:
Y = 4x – x^2
You can first use algebra techniques to plot this parabola, when x = 0, y=0. when x = 4, y=0. Maximum value occurs when x = 2, “-“ sign indicates the parabola is decreasing.
Have students graph and approximate the area under the curve. Area = 32/3.
You can also take the first derivative and set it equal to 0, 4 – 2x = 0, and the second derivative is negative, in a calculus class, to prove that the point at x = 2 is a max and concave down.
Have students integrate the parabola from 0 to 4 to determine the area under the curve, bounded by the x axis. Area = 32/3.
The axis of revolution is the equation:
Y=6.
Materials: SolidWorks, paper and pencil.
Procedures:
• Open the part, Parabola_Solid_in_Revolution.
• Click Sketch1 from the FeatureManager.
• Click Tools, Options, Section Properties to review the Area under the curve.
• Review the Centroid relative to the part origin. The Centroid is displayed in the Graphics window. Click Close.
• Click the axis of revolution at y=6.
• Click Revolve Feature.
• Enter 360 degrees. Click OK.
• Click Tools, Options, Mass Properties to view the Volume of the solid and the center of mass.
• Repeat for the Centroid_Solid_in_Revolution.
• Select the x axis as the axis of revolution.
• Review the volume.
• Review the center of mass.
• Change the material density. Display 4 decimal places in the Mass Properties option to calculate mass.
Assessment: On their own, have students create a parabola and revolve it around an axis. In algebra, plot the parabola to determine the total area. In Calculus I, integrate to determine area under the curve. In Calculus II, determine the center of mass and Volume of a Revolved solid.
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Resources Used: N/A
Copyrighted Materials:
• What materials did you employ from published sources?
• Ayres and Mendelson, Schaum’s Outline for Differential and Integral Calculus, McGraw-Hill, 1990
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