Putnam 1938 A1

A solid is bounded by two bases in the horizontal planes $z=h / 2$ and $z$ $=-h / 2$ , and by such a surface that the area of every section in a horizontal plane is given by a formula of the sort $\text { Area }=a_{0} z^{3}+a_{1} z^{2}+a_{2} z+a_{3}$ (where as special cases some of the coefficients may be 0 ). Show that the volume is given by the formula $V=\frac{1}{6} h\left[B_{1}+B_{2}+4 M\right]$ where $B_{1}$ and $B_{2}$ 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 $a_{0}=0$ .

Dataset text PutnamGAP. Quoted with source.

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