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Homework Help: Application with Shell Integration Method

  1. Jul 12, 2012 #1
    1. The problem statement, all variables and given/known data
    Use the shell method to find the volume of the solid obtained by rotating the region bounded by y=x2, y=0 and x=1 about the x-axis.

    2. Relevant equations
    lim Ʃ2∏RhΔw

    3. The attempt at a solution
    I realize this is hard to visualize without a graph. I uploaded how I split the graph.

    Solved for y and found y=0, 1.

    lim Ʃ2∏RhΔw
    =lim Ʃ2∏y(1-y)Δy // made it 1-y because y=x2.
    =lim Ʃ2∏(y-y2)Δy
    =2∏∫y-y2dy from 0->1
    =2∏[y2/2-y3/3] from 0->1

    Using Fundamental Theorem of Calculus, volume was found to be ∏/3. However this answer is incorrect. The answer is suppose to be ∏/5. I solved the question initially using the washer method and obtained the answer of ∏/5 but this question asks specifically to use shell method. I don't know how I can fix what I did wrong in this question. I assume I set the wrong h value. I also always get confused on what variable to use x or y, when taking into about if the rotation is about the x or y axis.

    Any help would be great thanks!

    Attached Files:

  2. jcsd
  3. Jul 12, 2012 #2


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    The quantity in parentheses is xright-xleft. You have (1-y). Now ##x_{right}=1## alright, but since ##y=x^2##, ## x_{left} =\sqrt y## so you should have ##(1-\sqrt y)## there. That will fix it.
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