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Predating the invention of calculus by almost 2000 years, archimedes found that the area under a parabolic arch is always 2 3 the area of the bounding rectangle.1 show that there is a similar relationship between the volume under a paraboloid and the volume of a can (cylinder) inside which the paraboloid sits.

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A paraboloid is a surface in space, a three-dimensional version of a parabola. It is assumed that the paraboloid opens downwards. So we can take it to have an equation in R^3 of the form z =BB−BB(x^2 + y^2), where you will want to fill in the BBs so that the piece of the paraboloid sitting above the xy-plane has radius R and height H.

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