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Homework Help: Volume using integration

  1. Sep 28, 2008 #1
    1. The problem statement, all variables and given/known data
    How do you find the volume of a region using integration?
    the region bounded by
    y = x^2
    y = 0
    x = 1
    x = 4

    and rotated about x = -1

    2. Relevant equations

    3. The attempt at a solution

    it doesn't seem to me you can just use the outer radius squared minus the inner radius squared b/c if you draw it out, the graph is bound by x = 1, so you get this shoe looking thing instead of the graph continuing to the origin, which then you could do.

    the outer radius would be 1+4 = 5
    the inner radus would be 1+sqrt(y)

    the limits of integration should be 0 to 16
  2. jcsd
  3. Sep 29, 2008 #2


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    Science Advisor
    Homework Helper

    Hi CrazyAmerican! :smile:
    (except of course that the inner radius is 1 for 0 ≤ y ≤ 1 :wink:)

    I don't understand the difficulty … shoe? origin? :confused:
  4. Sep 29, 2008 #3


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    Science Advisor

    Every cross section of the solid, parallel to the x-axis, is a circle with center (-1, y) and radius the distance from (-1,y) to (x, y) on the curve. Since y= x^2, x= sqrt(y) (we know this is the positive root because x is between 1 and 4) so the radius is sqrt(y)-(-1)= sqrt(y)+ 1. Since this starts at x= 1, the "inner radius" is 1+1= 2 (not 1+ sqrt(y) nor 1) and the outer radius is 1+ sqrt(2). When x= 1, y= 1, when x= 4, y= 16 so the integration should be done from y= 1 to y= 16.
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