Elementary Practical Mathematics With Numerous Exercises for the Use of Student
Elementary Practical Mathematics With Numerous Exercises for the Use of Student
John Perry
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For most of the simple solids, Simpson's rule gives the correct, and not merely an approximate answer. If A l and A z are the two parallel ends, and A. 2 is the area of the section parallel to the ends, midway between them, then the average section is The volume of the frustum of a cone or pyramid ; the volume of a sphere or of a paraboloid of revolution between two parallel cross- sections; or, indeed, of any ellipsoid and other surfaces of the second degree ; the volume of the frustum of a we...dge ; the volume of a prismoid these can all be computed exactly by Simpson's rule. For example, a whole sphere may be said to have A 1 = Q, A 3 = Q, and A 2 = irr 2, so that the average section is ^(0 + + 4:rr 2 ) or 47rr 2 /6. This multiplied by 2r is the volume = |^rT 3 . Definition of a Prismoid. Let there be two closed curves or irregular polygons on parallel planes, and let these be the ends of the prismoid. Imagine them joined by a surface which is made up of parts of cones or planes (a developable surface it is called).
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