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Math Help - Volume of bounded region

  1. #1
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    Volume of bounded region

    Let R denote the region enclosed by the y-axis, the line y = 1, and the curve
    y = sqrt(x). Compute the volume of the solid whose base is R and whose cross sections perpendicular to the y-axis are semicircles.

    I don't get it.. i did integral(pi*(y^2)^2)dy) from y=0 to y=1. i got pi/10
    The answer is supposed to be pi/40.
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  2. #2
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    Quote Originally Posted by bfpri View Post
    Let R denote the region enclosed by the y-axis, the line y = 1, and the curve
    y = sqrt(x). Compute the volume of the solid whose base is R and whose cross sections perpendicular to the y-axis are semicircles.

    I don't get it.. i did integral(pi*(y^2)^2)dy) from y=0 to y=1. i got pi/10
    The answer is supposed to be pi/40.
    Hi bfpri,

    y=\sqrt{x}

    x=y^2

    and all of these semicircles have diameter x, therefore radius (0.5)x.
    Their areas are half the area of a circle of the same radius.
    Therefore,

    \frac{1}{2}\int_{0}^1{\pi}r^2dy=\frac{\pi}{2}\int_  {0}^1\left(\frac{y^2}{2}\right)^2dy=\frac{\pi}{2}\  int_{0}^1\frac{y^4}{4}dy =\frac{\pi}{8}\frac{1^5}{5}=\frac{\pi}{40}
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  3. #3
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    thanks!
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