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Math Help - Volumes of Regions with Known Cross Sections

  1. #1
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    Volumes of Regions with Known Cross Sections

    Find the volume of the solid bounded by the curves y = x^2 and y = 8 - x^2 who cross sections are perpindicular to the x - axis with one side on the xy plane using semi circle cross sections

    Ok so i have to find total volume which is equal to the Rheiman sum of the the volume of the semi circle which is

    \frac{1}{2}\pi*r^2*\delta thickness

    and where delta thickness is the change in x?
    and the radius of the semi circle is 1/2 the height between y = 8 - x^2 and y = x^2
    which gives

    \frac{8-2x^2}{2}

    4-x^2

    then you will have the intergral from x = -2 to x = 2 because this is where they intersect

    \frac{\pi}{2}\int{4-x^2}^2

    For some reason I dont think im accurately doing this
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  2. #2
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    Quote Originally Posted by RockHard View Post
    Find the volume of the solid bounded by the curves y = x^2 and y = 8 - x^2 who cross sections are perpindicular to the x - axis with one side on the xy plane using semi circle cross sections

    Ok so i have to find total volume which is equal to the Rheiman sum of the the volume of the semi circle which is

    \frac{1}{2}\pi*r^2*\delta thickness

    and where delta thickness is the change in x?
    Yes, because your cross sections are "perpendicular to the x-axis", the thickness, perpendicular to the area, is measured along the x-axis.

    and the radius of the semi circle is 1/2 the height between y = 8 - x^2 and y = x^2
    which gives

    \frac{8-2x^2}{2}

    4-x^2

    then you will have the intergral from x = -2 to x = 2 because this is where they intersect

    \frac{\pi}{2}\int{4-x^2}^2

    For some reason I dont think im accurately doing this
    Two errors:
    You left out the "thickness" which becomes "dx" in the limit. NEVER write an integral without the "dx"!
    Also, it is the entire polynomial that is squared. \frac{\pi}{2}\int \left(4- x^2\right)^2 dx.

    Now, what are the limits of integration?
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    The limits of integration are lower limit = -2 and upper limit = 2 because thats where the two graphs intersect?
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  5. #5
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    ...again..bump
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