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Calculus: Area and volume

Please see the attached file for full problem description.

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1. A solid is bounded by two bases in the horizontal planes z = h/2 and z = -1/2, and by such a surface that the area of every section in a horizontal plane is given by a formula of the sort

Area = a0 z3 + a1 z2 + a2 z + a3

(where as special cases some of the coefficients may be 0). Show that the volume is given by the formula

V = 1/6?h [B1 +B2 + 4M],

where B1 and B2 are the areas of the bases, and M is the area of the middle horizontal section. Show that the formulas for the volume of a cone and of a sphere can be included in this formula when a0 = 0

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Solution Summary

The function of area is given and the volume is calculated.

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