Calculus: Analytic Geometry and Calculus, with Vectors

4.3 Properties of integrals 225

Theorem 4.341 If k is a constant, then

f-22

k dt = k(x2 - xl).

If a < b, if f is integrableover a <- x 5 b, and if

m 5 f(x) < M (a c x< b)

m(b - a) <

fb

f(t) dt < M(b- a)

lab

mob l a f(t)dt<M.

Theorem 4.343 If f is integrable over a < x <- b, then so
function having values Jf (x) and

f x2 f (x) dx fx2 l f(x)I dx

also is the

whenever xl and x2 lie between a and b.

The next theorem is not so obvious, and it is so important that we shall

discuss it and prove it. Much of the theory and many of the applications

of the calculus involve relations between derivatives and integrals.

Theorems which give information about derivatives of integrals or inte-

grals of derivatives are called fundamental theorems of the calculus. The

following theorem is one of the best of these. It has very many applica-

tions and shows, among other things, that if f is continuous over a 5 x S

b, then there exists a function F for which F(x) = f(x) when a S x S b.

In fact, it shows that if f is continuous, then the Riemann integral in

(4.351) is an "indefinite integral" of f.

Theorem 4.35 If f is integrable over a < x <_ b, then the function F

defined by

(4.351) F(x) =

fas

f(t) dt

is continuous over a <- x 5 b and

(4.352) F(x) = f(x)

for each x for which f is continuous.

To start the proof, we observe that if x and x + Ox both lie in the inter-

val, then

(4.353) F(x + Ox) - F(x) =fax+ex f(t)dt - faxf(t) dt =

f70+' f(t)

dt.

To prove continuity of F, we use Theorem 4.26 to see that f must be