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AP®MathsCollege BoardCalculus ABStudy GuidesUnit 8: Applications of IntegrationVolumes with Cross SectionsVolumes from Areas of Known Cross Sections

Volumes from Areas of Known Cross Sections (College Board AP® Calculus AB)

Study Guide

Roger B

Written by: Roger B

Reviewed by: Dan Finlay

Volumes from areas of known cross sections

How do I find the volume of a solid with known cross sections?

  • If the area of the cross section of a solid

    • can be expressed as a function of bold italic x

      • i.e. as a function A that may be written in the form A open parentheses x close parentheses

    • and if A open parentheses x close parentheses is continuous on the interval 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

  • For example the cross-sectional area of a solid is given by A open parentheses x close parentheses equals fraction numerator 1 over denominator x plus 1 end fraction comma space space 0 less or equal than x less or equal than 3

    • The volume of the solid between x equals 0 and x equals 3 is

      • Volume equals integral subscript 0 superscript 3 fraction numerator 1 over denominator x plus 1 end fraction space d x equals open square brackets ln open parentheses x plus 1 close parentheses close square brackets subscript 0 superscript 3 equals ln 4 minus ln 1 equals ln 4 minus 0 equals ln 4

  • This method of finding volumes uses the idea of a definite integral as calculating an accumulation of change

    • A open parentheses x close parentheses times increment x is the volume of a solid with cross-sectional area A open parentheses x close parentheses and length increment x

    • A open parentheses x close parentheses space d x is the limit of this volume element as increment x rightwards arrow 0

    • The integral integral subscript a superscript b A open parentheses x close parentheses space d x sums up all these infinitesimal volume elements between x equals a and x equals b

Worked Example

The area, in square feet, of the the horizontal cross section of a water tank at height h feet is modeled by the function f given by f open parentheses h close parentheses equals 50 over e to the power of h. The tank has a height of 10 feet.

Based on this model, find the volume of the tank. Indicate units of measure.

Answer:

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

table row Volume equals cell integral subscript 0 superscript 10 50 over e to the power of h space d h end cell row blank equals cell 50 integral subscript 0 superscript 10 e to the power of negative h end exponent space d h end cell row blank equals cell 50 open square brackets negative e to the power of negative h end exponent close square brackets subscript 0 superscript 10 end cell row blank equals cell 50 open parentheses negative e to the power of negative open parentheses 10 close parentheses end exponent minus open parentheses negative e to the power of negative open parentheses 0 close parentheses end exponent close parentheses close parentheses end cell row blank equals cell 50 open parentheses 1 minus e to the power of negative 10 end exponent close parentheses end cell row blank equals cell 49.9977... end cell end table

Don't forget to give the units in the final answer

Area has units of feet squared, and height has area of feet, so the units of volume are feet cubed (or cubic feet)

49.998 cubic feet (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.