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Math Help - volume of solid hemisphere

  1. #1
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    Post volume of solid hemisphere

    Estimate the volume of a solid hemisphere of radius 3, imagining the axis of symmetry to be the x-axis. Partition the interval [0,3] into six subintervals of equal length and approximate the solid with cylinders based on the circular cross sections of the hemisphere perpendicular to the x-axis at the subintervals' left endpoints.

    How would I start this problem?
    Any help is appreciated!
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  2. #2
    Senior Member apcalculus's Avatar
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    Quote Originally Posted by live_laugh_luv27 View Post
    Estimate the volume of a solid hemisphere of radius 3, imagining the axis of symmetry to be the x-axis. Partition the interval [0,3] into six subintervals of equal length and approximate the solid with cylinders based on the circular cross sections of the hemisphere perpendicular to the x-axis at the subintervals' left endpoints.

    How would I start this problem?
    Any help is appreciated!

    So we're using Left Hand Sums.

    First find the height of each cylinder.

    \Delta x = \frac{b-a}{n} = \frac{3 - 0}{6} = 0.5<br />

    It's six subintervals, so we'll have six cylinders.

    Note that the height function is the upper half of the circle of radius 3:

    y=\sqrt{9-x^2}

    The radii for the six cylinders are the y-values of the left end points:

    [0, 1/2]
    [1/2, 2/2]
    [2/2, 3/2]
    [3/2, 4/2]
    [4/2, 5/2]
    [5/2, 6/2]

    The Volume formula is

    V_i = \pi r^2 h = \pi y^2 \Delta x

    for example:

    V_1 = \pi (f(o))^2 0.5 = \pi 9 * 0.5 = \frac{9 \pi}{2}
    V_2 = \pi (f(1/2))^2 0.5 = \pi \frac{35}{4} * 0.5 = \frac{35 \pi}{8}
    ....

    Compute the remaining cylinders and add them up to get the approximation.

    Good luck!!
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