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Math Help - Volume by cross-sections (possibly misleading)

  1. #1
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    Volume by cross-sections (possibly misleading)

    The basis of a solid is the circle . Every cross-section perpendicular to the x axis is a rectangle whose height is twice the distance from the origin of the circle to the plane of the cross-section. Find the volume of the solid.

    Note: I don't know what the problem means by "origin of the circle". I assumed it means the center of the circle, (8,0), but I think it could mean instead the origin (0,0).

    This is my reasoning: the distance mentioned is , so the height of a cross-section is , assuming . The length of a cross-section is , so the cross-sectional area is . Integrate this from x = 0 to x = 8, multiply by 2 to cover the other half of the solid and you find . But the answer is supposed to be or something like that.

    Is this right? Is the problem misleading? Any help would be greatly appreciated.
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  2. #2
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    (x-8)^2 + y^2 = 8^2

    horizontal distance from the origin (0,0) to a representative cross-section is x.

    area of a representative cross-section ...

    A = 2x \cdot 2y = 4xy = 4x\sqrt{64-(x-8)^2}

    V = \int_0^{16} 4x\sqrt{64-(x-8)^2} \, dx = 1024\pi
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