4.9 Simpson and other approximations to integrals 2839 An application of (2) of Problem 8 gives the famous old prismoidal formula(3) V= 6 [IB1I+4IMI+IB21]for volumes of solids. To investigate this matter, put xo= a and x2 = a + H in
(2) to obtain(4)Iaa+Flf(x)dx= 6 [f(a)+4f(a+?)+f(a+H)]
If a reasonably decent solid has bases in the planes having the equations x = a
and x = a + H, and if for each x' for which a < x' < a + H the plane having
the equation x = x' intersects the solid in a plane region having area f(x'), then
the left member of (4) is the volume Y of the solid. The quantity in brackets in
(4) is the sum of the area IB1i of one base B1, the area IB21 of the other base B2,
and four times the area IMI of the section M midway between the two bases.
Thus the formula (3) is correct when the solid has volume Y equal to the left
member of (4) and f(x) has the form
f(x) = K1x3 + K2x2 + K3x + K4.Nearly everyone acquires substantial respect for the prismoidal formula when it
is discovered that the formula yields the correct formula for the volume of a
spherical ball of radius a. In this case H = 2a, the bases are points having area 0,
and the midsection M is an equatorial disk having area Tra2.
10 While the matter cannot be fully explored in a course in elementary
calculus, we can know that persons who study Lebesgue measure and integration
may learn that E3 contains sets much queerer than those considered in this book.
It can happen that each plane section perpendicular to the x axis is a square of
unit area so that (in the context of Problem 9) f (x) = 1 when 0 <= x < 1, but,
nevertheless, the squares are so heterogeneously scattered that the set fails to
possess a volume. For such queer sets the prismoidal formula is invalid because
the left member of (4) of Problem 9 is not the volume of the set. Experts in the
theory of measure can have sympathy for students of solid geometry who are a
bit mystified by the "Cavalieri theorem." This "theorem" says that two sets in
E3 have equal volumes if they have parallel bases and equal altitudes, and if each
plane parallel to the bases intersects the two sets in two plane regions having
equal areas. The queer sets which we have mentioned show that the "theorem"
is false. Appendix 2 at the end of this book shows how we can reconcile ourselves
to these matters. Some of us will learn more about these things than others, but
we can all know that there is much to be learned.