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Math Help - Finding out the amount of volume through integration

  1. #1
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    Finding out the amount of volume through integration

    A right circular cone has height 4 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 − x^2 . 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?
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  2. #2
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    So you have to integrate in two sections - the cone and the paraboloid. The equation for the cone is y=2x and y goes from 0 to 4. The equation for the paraboloid is y=8-x^2 with y going from 4 to 8. The element of volume is a cylinder of radius x and height dy, so its volume is \pi{x}^2\ dy. The volume is given by:

    V=\int\pi{x}^2\ dy=\int_0^4\pi\left(\frac{y}{2}\right)^2\ dy+\int_4^8\pi(8-y)\ dy

    - Hollywood
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