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AP®MathsCollege BoardCalculus ABStudy GuidesUnit 8: Applications of IntegrationVolumes of RevolutionWasher Method Around the y-Axis

Washer Method Around the y-Axis (College Board AP® Calculus AB)

Study Guide

Roger B

Written by: Roger B

Reviewed by: Dan Finlay

Volume with washer method revolving around the y-axis

How can I use the washer method to calculate a volume of revolution around the y-axis?

  • This is very similar to the washer method for volumes of revolution around the x-axis

    • Use this method when there is a gap between the region to be rotated and the y-axis

  • Let f and g be continuous functions of y such that open vertical bar f open parentheses y close parentheses close vertical bar less than open vertical bar g open parentheses y close parentheses close vertical bar on the interval open square brackets a comma space b close square brackets

    • I.e. f open parentheses y close parentheses is closer to the y-axis than g open parentheses y close parentheses is on that interval

  • If the region bounded by

    • the curves x subscript 1 equals f open parentheses y close parentheses and x subscript 2 equals g open parentheses y close parentheses

    • between y equals a and y equals b

  • is rotated 2 pi radians left parenthesis 360 degree right parenthesis around the y-axis, then the volume of revolution is

    •  V equals integral subscript a superscript b pi open parentheses x subscript 2 squared minus x subscript 1 squared close parentheses space italic d y equals pi integral subscript a superscript b open parentheses x subscript 2 squared minus x subscript 1 squared close parentheses space italic d y

      • Note that x subscript 1 and x subscript 2 are both functions of y

      • If the functions are given as functions of x

        • e.g y equals p open parentheses x close parentheses and y equals q open parentheses x close parentheses

      • then you will need to rewrite them as functions of y

        • See the Worked Example

      • Also note that the integration is done with respect to y

  • Make sure that x subscript 2 is the curve further away from the y-axis

    • and x subscript 1 is the curve closer to the y-axis

      • If the curves 'swap places' over the interval

        • then split the calculation into separate integrals

  • If y equals a and y equals b are not stated in a question, these boundaries could involve

    • the x-axis (y equals 0)

    • and/or point(s) of intersection of the two curves

Examiner Tips and Tricks

Be careful not to confuse open parentheses x subscript 2 squared minus x subscript 1 squared close parentheses with open parentheses x subscript 2 minus x subscript 1 close parentheses squared

  • These are not equal!

    • open parentheses x subscript 2 minus x subscript 1 close parentheses squared equals x subscript 2 squared minus 2 x subscript 1 x subscript 2 plus x subscript 1 squared

Worked Example

Let R be the region enclosed by the graphs of f open parentheses x close parentheses equals 1 fourth x squared and g open parentheses x close parentheses equals x, as shown in the figure below.

Graph showing a shaded region R enclosed by the curves y=x and y=x^2/4

Find the volume of the solid generated when R is rotated about the y-axis.

Answer:

Use V equals pi integral subscript a superscript b open parentheses x subscript 2 squared minus x subscript 1 squared close parentheses space italic d y

First rewrite the functions as functions of y

table row y equals cell f open parentheses x close parentheses end cell row y equals cell 1 fourth x squared end cell row cell x squared end cell equals cell 4 y end cell row x equals cell 2 square root of y end cell end table

Note that x equals 2 square root of y is used instead of x equals negative 2 square root of y because it can be seen from the graph that the x values of the relevant part of the curve are positive

table row y equals cell g open parentheses x close parentheses end cell row y equals x row x equals y end table

g open parentheses x close parentheses equals x is the function closest to the y-axis, so use x subscript 1 equals y and x subscript 2 equals 2 square root of y

To find a and b, solve x subscript 1 open parentheses y close parentheses equals x subscript 2 open parentheses y close parentheses to find the y-coordinates of the points of intersection of the two curves

table row y equals cell 2 square root of y end cell row cell y squared end cell equals cell 4 y end cell row cell y squared minus 4 y end cell equals 0 row cell y open parentheses y minus 4 close parentheses end cell equals 0 end table

table row y equals cell 0 space space or space space y equals 4 end cell end table

So a equals 0 and b equals 4

Set up and solve the integral

table row V equals cell pi integral subscript 0 superscript 4 open parentheses open parentheses 2 square root of y close parentheses squared minus open parentheses y close parentheses squared close parentheses space italic d y end cell row blank equals cell pi integral subscript 0 superscript 4 open parentheses 4 y minus y squared close parentheses space italic d y end cell row blank equals cell pi open square brackets 2 y squared minus 1 third y cubed close square brackets subscript 0 superscript 4 end cell row blank equals cell pi open parentheses open parentheses 2 open parentheses 4 close parentheses squared minus 1 third open parentheses 4 close parentheses cubed close parentheses minus 0 close parentheses end cell row blank equals cell fraction numerator 32 pi over denominator 3 end fraction end cell row blank equals cell 33.510321... end cell end table

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

33.510 units3 (to 3 decimal places)

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    Previous:Washer Method Around the x-AxisNext:Washer Method Around Other Axes
    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.