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

Semicircles 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 semicircles

How can I find the volume of a solid with a semicircular 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

  • Remember that the area of a circle is pi cross times radius squared

  • So the area of a semicircle with radius r is

    • Area equals 1 half cross times pi r squared

Worked Example

Let R be the triangular region with vertices open parentheses 0 comma space 0 close parentheses, open parentheses 0 comma space 2 close parentheses and open parentheses 4 comma space 0 close parentheses, as shown in the figure below.

A graph of a shaded region labeled R, where R is the triangle with vertices (0, 0), (0, 2) and (4, 0)

Region R is the base of a solid. For the solid, at each x the cross section perpendicular to the x-axis is a semicircle. Find the volume of the solid.

Answer:

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

To define A open parentheses x close parentheses, first find the equation of the line through points open parentheses 0 comma space 2 close parentheses and open parentheses 4 comma space 0 close parentheses

gradient equals fraction numerator 0 minus 2 over denominator 4 minus 0 end fraction equals negative 1 half

table row cell y minus 0 end cell equals cell negative 1 half open parentheses x minus 4 close parentheses end cell row y equals cell 2 minus 1 half x end cell end table

That's the diameter of each semicircle; to find the radius divide by two

radius equals fraction numerator open parentheses 2 minus 1 half x close parentheses over denominator 2 end fraction equals 1 minus 1 fourth x

At each x the cross-sectional area is 1 half pi r squared

table row cell A open parentheses x close parentheses end cell equals cell 1 half pi open parentheses 1 minus 1 fourth x close parentheses squared end cell row blank equals cell 1 half pi open parentheses 1 minus 1 half x plus 1 over 16 x squared close parentheses end cell row blank equals cell pi over 32 open parentheses 16 minus 8 x plus x squared close parentheses end cell end table

Now the volume integral can be used

table row Volume equals cell integral subscript 0 superscript 4 pi over 32 open parentheses 16 minus 8 x plus x squared close parentheses space d x end cell row blank equals cell pi over 32 integral subscript 0 superscript 4 open parentheses 16 minus 8 x plus x squared close parentheses space d x end cell row blank equals cell pi over 32 open square brackets 16 x minus 4 x squared plus 1 third x cubed close square brackets subscript 0 superscript 4 end cell row blank equals cell pi over 32 open parentheses open parentheses 16 open parentheses 4 close parentheses minus 4 open parentheses 4 close parentheses squared plus 1 third open parentheses 4 close parentheses cubed close parentheses minus open parentheses 0 close parentheses close parentheses end cell row blank equals cell pi over 32 open parentheses 64 minus 64 plus 64 over 3 close parentheses end cell row blank equals cell fraction numerator 2 pi over denominator 3 end fraction end cell row blank equals cell 2.094395...... end cell end table

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

2.094 units3 (to 3 decimal places)

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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.