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AP®MathsCollege BoardCalculus ABStudy GuidesUnit 8: Applications of IntegrationVolumes with Cross SectionsRectangles as Cross Sections

Rectangles as Cross Sections (College Board AP® Calculus AB)

Study Guide

Roger B

Written by: Roger B

Reviewed by: Dan Finlay

Volumes with cross sections as rectangles

How can I find the volume of a solid with a rectangular cross section?

  • Use the basic concept

    • If the area of the cross section of a solid is given by A open parentheses x close parentheses

      • and A open parentheses x close parentheses is continuous on open square brackets a comma space b close square brackets

    • Then the volume of the corresponding solid from x equals a to x equals b is

      • Volume equals integral subscript a superscript b A open parentheses x close parentheses space d x

  • You may need to create the cross sectional area function A open parentheses x close parentheses based on information provided

    • For example A open parentheses x close parentheses may depend on the values of another function (or functions) given to you in the question

Worked Example

Let R be the region enclosed by the graph of f open parentheses x close parentheses equals 54 minus 2 x cubed, the x- and y-axes, and the vertical line x equals 2, as shown in the figure below.

Graph shows a shaded region R enclosed by the x-axis, y-axis, the line x = 2, and the curve y=54+2x^3.

Region R is the base of a solid. For the solid, at each x the cross section perpendicular to the x-axis is a rectangle with height h open parentheses x close parentheses equals 1 plus x. Find the volume of the solid.

Answer:

Use Volume equals integral subscript a superscript b A open parentheses x close parentheses space d x

At each x, the cross-sectional area is A open parentheses x close parentheses equals f open parentheses x close parentheses times h open parentheses x close parentheses equals open parentheses 54 minus 2 x cubed close parentheses open parentheses 1 plus x close parentheses

table row Volume equals cell integral subscript 0 superscript 2 open parentheses 54 minus 2 x cubed close parentheses open parentheses 1 plus x close parentheses space d x end cell row blank equals cell integral subscript 0 superscript 2 open parentheses 54 plus 54 x minus 2 x cubed minus 2 x to the power of 4 close parentheses space d x end cell row blank equals cell open square brackets 54 x plus 27 x squared minus 1 half x to the power of 4 minus 2 over 5 x to the power of 5 close square brackets subscript 0 superscript 2 end cell row blank equals cell open parentheses 54 open parentheses 2 close parentheses plus 27 open parentheses 2 close parentheses squared minus 1 half open parentheses 2 close parentheses to the power of 4 minus 2 over 5 open parentheses 2 close parentheses to the power of 5 close parentheses minus open parentheses 0 close parentheses end cell row blank equals cell 108 plus 108 minus 8 minus 64 over 5 end cell row blank equals cell 976 over 5 end cell row blank equals cell 195.2 end cell end table

The question doesn't specify units, so the units of volume will be units cubed

195.2 units3

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    Author
    Reviewer
    Roger B

    Author: Roger B

    Expertise: Maths

    Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.

    Dan Finlay

    Author: Dan Finlay

    Expertise: Maths Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.