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How to find the volume of a square with function-based side?

  1. Dec 7, 2015 #1
    1. The problem statement, all variables and given/known data
    "[Find the volume of a] solid whose base is the region bounded by the curves (y = x2) and y = 2 - x2 and whose cross sections through the solid perpendicular to the x-axis are squares."

    2. Relevant equations
    A(f(x)) = f(x)2
    V = ∫(A(f(x))dx
    Image of problem: http://i.imgur.com/6FtsBzO.png
    Answer (from back of book): V = 64/15.


    3. The attempt at a solution
    y = (2 - x2) - (x2) = 2 - 2x2
    A(y) = (2 - 2x2)2 = 4x4 - 4x2 + 4
    V = ∫A(f(x)) = (4/5)x5 - (4/3)x3 + 4x on [-1,1]
    [(4/5) - (4/3) + 4] = [(12 - 20 + 60)/(15)] - [(-12 + 20 - 60)/(15)] = (104/15) ≠ (64/15)

    I think I'm doing this right, but I'm not getting the right numbers. Could someone help me with this?
     
  2. jcsd
  3. Dec 7, 2015 #2

    Student100

    User Avatar
    Education Advisor
    Gold Member

    Check your setup math, (x^2-(2-x^2))^2 would be the region you want to look at.

    You have two fours, I think you just made a math mistake. =)
     
  4. Dec 7, 2015 #3
    You're right.

    A(y) = (2 - 2x2)2 = 4x4 - 8x2 + 4
    V = ∫A(f(x)) = (4/5)x5 - (8/3)x3 + 4x on [-1,1]
    [(4/5) - (8/3) + (4)] - [(-4/5) + (8/3) - (4)] = 2(12 - 40 + 60)/(15) = 2(32/15) = 64/15

    Thanks!
     
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