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Homework Help: Volume generated by rotating a region about a specified line

  1. Aug 29, 2010 #1
    1. The problem statement, all variables and given/known data
    Find the volume generated by rotating the given region about the specified line

    The region, R2, is y = x^1/2 (y = sqrt(x) )
    The line, AB, is x = 1

    So it's y = x^1/2 rotated about x = 1

    I've solved this equation easily numerous times using cylindrical shells and just cylinders with respect to dy and I get 8pi/15 every time, while the book claims it's 7pi/15

    2. Relevant equations
    Cylindrical shells method:
    V(x) = 2pi r h dr so V(x) = 2pi(1 - x)(x^1/2)dx

    Or, with cylinders...
    If y = x^1/2 then x = y^2
    So if V(x) = pi r^2 dr then
    V(x) = pi (1-x)^2 dx Substitute y for x you get V(y) = pi {1-(y^2)}^2 dy

    3. The attempt at a solution

    Cylindrical shell style...(integral where b = 1, a = 0)
    V(x) = 2pi(1-x)(x^1/2) dx
    2pi |x^1/2 - x^3/2 dx
    (x^3/2)/(3/2) - (x^5/2)/(5/2) = (2x^3/2)/3 - (2x^5/2)/5
    (2(1)^3/2)/3 - (2(1)^5/2)/5 = 2/3 - 2/5 = 10/15 - 6/15 = (4/15)*2pi = 8pi/15 which is incorrect, the book says it's 7pi/15

    Disk method...(integral where b = 1, a = 0)
    V(y) = pi {1-(y^2)}^2 dy
    (1-y^2)(1-y^2) = 1 - y^2 - y^2 + y^4 = 1 - 2y^2 + y^4 Integral it....
    pi| y - 2y^3/3 + y^5/5 = 1 - (2(1)^3)/3 + ((1)^5)/5 = 15/15 - 10/15 + 3/15 = 8/15*pi = 8pi/15 which is incorrect, the book says it's 7pi/15

    Sorry for the messy work, any help would be appreciated! Maybe it's not the integral from 0 to 1? Perhaps 2 to 1? I'm going to try that next. Also the book may be wrong, I certainly hope that if that's the case that the grader knows that as well!

    EDIT: Alright, I'm pretty sure the book is wrong on this one. I've been doing a group of problems concerning the same 3 regions divided by the same 2 functions rotated about the same lines. One of them is the region between R1 and R2. I found R1 (it's the region below y = x^3, so when rotated about AB (x =1 ) it's volume was pi/10 and the book agrees with that. R2, my problem that I posted here, when used as what the book says it is, 7pi/15, I get the volume between these regions as 11pi/30 (work: 70pi/150 - 15pi/150 = 55pi/150 = 11pi/30) while the book says it's 13pi/30. Well guess what? If I use my answer 8pi/15 instead of 7pi/15, I get the correct answer. So I'm pretty sure that proves that the book screwed up in this case. But just to be sure, tell me what you guys get. Thanks!
    Last edited: Aug 29, 2010
  2. jcsd
  3. Aug 29, 2010 #2


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    Science Advisor
    Homework Helper

    I get 8*pi/15.
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