Calculus: Analytic Geometry and Calculus, with Vectors

224 Integrals

4.3 Properties of integrals In what follows, all integrals bearing

limits of integration are Riemann integrals. They are limits of Riemann
sums, and it could be expected that, except for cases in which the inte-
grands are step functions, it must be impossible to obtain their exact
values and it must be difficult to obtain reasonably good approximations
to them. It turns out, however, that there is a calculus, an invention of
Newton and Leibniz, by which exact values of very many of the most
important integrals can be calculated very quickly. Dictionaries tell us
that a calculus is "a method of computation." The particular calculus
that appears at the end of this section was found to be so overwhelmingly
important that it came to be known as "the calculus." This calculus
enables us, for example, to evaluate the integral in the formula

(4.31) fx2dx^3 2 7

=^3

by writing nothing more than this. Meanings of words have evolved in

such a way that we now consider "calculus" or "the calculus" to be a

name assigned to a part of mathematics involving derivatives and

integrals. t

For making calculations involving integrals, we often need the results

set forth in the following theorems. Proofs of these theorems may be

omitted; these theorems are rather simple consequences of Theorem 4.26

and properties of Riemann sums and their limits.

Theorem 4.32 If f is integrable over an interval containing a, b, and c,

then

f cf(t) dt + fbf(t) dt = f b f(t) dt.

Theorem 4.33 If f and g are integrable over a < x <_ b and .4 and B

are constants, then

fzxz

[.1f(t) + Bg(t)l dt = A

fx2,

f(t) dt + B

fyxs

g(t) dt

whenever xi and x2 lie in the interval a < x 5 b.

Theorem 4.34 If a < b, if fl, f2, fa are integrable over a < x < b, and if

fl(x) < f2(X) < f3(x) (a s x s b)

then

f ab f l(t) dt <

fA

f2(t) dt c f afa(t) dt.

t Historians who claim that Archimedes knew calculus do not always point out that the

knowledge was attained posthumously when the meaning of "calculus" changed. Com-

plete misunderstanding of this matter can serve as a basis for the absurd contention that

Newton and Leibniz merely rediscovered inventions of Archimedes.