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Math Help - Volume

  1. #1
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    Volume

    Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. y=2x^2, 0<x<2, y=8; about the y-axis.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by asnxbbyx113 View Post
    Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. y=2x^2, 0<x<2, y=8; about the y-axis.
    The area of a disk of radius x=sqrt(y/2) is pi x^2= pi y/2. The volume you
    seek is the integral of this area from y=0 to y=8 or:

    V = integral(y=0:8) pi y/2 dy = pi/2 integral(y=0:8) y dy = pi/2 [8^2/2 - 0] = 16 pi

    RonL
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  3. #3
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    Hello, asnxbbyx113!

    This can be done with cylindrical shells, too.


    Find the volume of the solid obtained by rotating the region bounded by:
    . . y = 2x, 0 < x < 2, y = 8; about the y-axis
    Code:
              |
             8+ - - - - *
              |:::::::::
              |::::::::*
              |::::::*
              |:::*
          ----*----------+---
              |          2
    . . V . = . xy dx

    . . . . . . . . . . . . . . .
    2
    We have: .V .= .
    x(8 - 2x) dx . = . 16π
    . . . . . . . . . . . . . .
    0

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