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Math Help - Volume of a solid

  1. #1
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    Volume of a solid

    A solid is formed when the area bounded by y=8-0.5x^2, x=0 and y=0 is rotated about the y axis. Determine the volume of the solid.

    How do I go about solving this?
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  2. #2
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    Quote Originally Posted by jigger1 View Post
    A solid is formed when the area bounded by y=8-0.5x^2, x=0 and y=0 is rotated about the y axis. Determine the volume of the solid.

    How do I go about solving this?
    I would use cylindrical shells to solve this problem. The height of each shell will be 8-0.5x^2. The form for cylindrical shells is \int_{a}^{b} 2\pi{r}{h(r)}dr. In this case, you will integrate \int_{0}^{4} 2\pi{x}{(8-0.5x^2)}dx.
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  3. #3
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    y = 8 - \frac{1}{2}x^{2} intersects the x-axis at x = 4.

    So by method of cylindrical shells, we have:
    V = \int_{0}^{4} A(x) dx where A(x) is the surface area of the cylinder
    V = \int_{0}^{4} 2\pi r h

    If you make a quick sketch of the graph, you'll see it's an inverted parabola. If you imagine drawing one of the cylinders from the y-axis, you'll see that the radius is the horizontal distance from the y-axis to the curve which is just simply x. The height of the cylinder is the distance from the x-axis to the curve (which is simply the curve it self).

    So:
    V = 2\pi \int_{0}^{4} x \left(8 - \frac{1}{2}x^{2}\right)dx

    Edit: Ok too slow xD
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  4. #4
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    Thanks icemanfan. That gets me started.
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  5. #5
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    To o_O: But your explanation is better than mine
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  6. #6
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    Can any one assist/explain how to solve from here?
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  7. #7
    Behold, the power of SARDINES!
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    Quote Originally Posted by jigger1 View Post
    Can any one assist/explain how to solve from here?
    Evauate the integral

    V = 2\pi \int_{0}^{4} x \left(8 - \frac{1}{2}x^{2}\right)dx

    V = 2\pi \int_{0}^{4} \left(8x - \frac{1}{2}x^{3}\right)dx=2\pi\left( 4x^2-\frac{1}{8}x^4\right)|_0^{4}

    2\pi(64-32)=64\pi
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  8. #8
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    We could also use washers. Solve the given for x and we get

    x=\sqrt{16-2y}

    {\pi}\int_{0}^{8}(16-2y)dy
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