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Homework Help: Circular cone volume through integration

  1. May 12, 2010 #1
    1. The problem statement, all variables and given/known data
    A right circular cone has height 6 cm and base radius 2. It is over-filled with ice cream,
    in the usual way. Place the cone so its vertex is at the origin, and its axis lies along the
    positive y–axis, and take the cross-section containing the x–axis. The top of this crosssection
    is a piece of the parabola y = 8 − x2 . The whole filled ice-cream cone is obtained
    by rotating this cross-section about the y–axis.
    What is the volume of the ice cream?

    2. Relevant equations

    3. The attempt at a solution
    So for I have
    x2 = 8-y
    v = pi . integral((8-y)dx) from 0 to 8

    I am not sure if I am on the right path though.
    Many thanks,
  2. jcsd
  3. May 12, 2010 #2


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    Science Advisor
    Homework Helper

    Welcome to PF!

    Hi orangesun! Welcome to PF! :smile:

    (have an integral: ∫ and a pi: π :wink:)
    Yes, that's the right path for the curved part of the ice-cream.

    (except it isn't dx, it's dy … each horizontal slice is a disc of area πx2 and height dy)

    Now you need to decide on the limits of integration (for y), and then add the volume of the cone part. :smile:

    (btw, is your parabola correct? it doesn't seem to meet the top of the cone … and we wouldn't want to lose any ice-cream! :redface:)
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