fundamental theorem of calculus 211

dA

––

dx

The fundamental theorem of arithmetic is used to

prove the irrationality of roots of integers. Precisely,

one can show that the square root of a positive integer

ais a rational number if, and only if, ais a perfect

square (that is, a= b2for some whole number b). This

is established as follows:

Suppose that √

–

ais a fraction. Then √

–

a= n/m

for some whole numbers nand m:

Squaring and simplifying gives am2= n2. The

prime factorization of ngives a prime factoriza-

tion of n2with the property that every prime

that appears does so an even number of times.

As this is the prime factorization of am2, and

the primes that appear in m2already do so an

even number of times, it follows that the primes

that appear in the factorization of ado so an

even number of times as well. Selecting one

prime from each pair of primes that appears

produces a whole number bwith the property

b2= a. Thus the assumption that √

–

ais rational

leads us to conclude that ais a perfect square.

Consequently, since the number 2, for instance, is not a

perfect square, it cannot be the case that √

–

2 is rational.

Thus √

–

2 must be an irrational number. (A similar argu-

ment establishes that n

√

–

ais rational if, and only if, ais a

perfect nth power.)

See also

COMMON FACTOR

;

COMMON MULTIPLE

;

FACTOR

.

fundamental theorem of calculus The theory of

CALCULUS

develops methods for calculating two impor-

tant quantities associated with curves, namely, the

slopes of tangent lines to curves and the areas of

regions bounded by curves. Although isolated problems

dealing with these issues have been studied since the

time of antiquity (see

HISTORY OF CALCULUS

essay), no

unified approach or technique for solving them was

developed for a very long time. A breakthrough came

in the 1670s when G

OTTFRIED

W

ILHELM

L

EIBNIZ

in

Germany and S

IR

I

SAAC

N

EWTON

in England indepen-

dently discovered a fundamental inverse relationship

between the tangent problem (differentiation) and the

area problem (integration). Their result, today known

as the fundamental theorem of calculus, binds together

the two parts of the subject and is no doubt the most

important single fact in the whole of mathematics.

Leibniz and Newton each recognized the importance of

the result, developed the ideas that follow from it, and

applied the consequent results to solve problems in sci-

ence and geometry with what can only be described as

spectacular success. Their discovery of the fundamental

theorem is, in essence, the discovery of calculus.

The theorem has two parts. Let y= f(x) be a con-

tinuous curve. If A(x) represents the area under the

curve from position ato position x, then the

RATE OF

CHANGE

of area, , is given by:

Noting that the area under the curve between positions

xand x+ hcan be approximated as a rectangle of

width hand height f(x),we have A(x+ h) – A(x) ≈

f(x)·h. Thus . The details of

the argument can be made rigorous to prove:

The rate of change of the area under a curve

y= f(x) with respect to xis f(x).

Loosely speaking, “taking the

DERIVATIVE

of an

INTE

-

GRAL

returns the original function.” In particular, the

function A(x) is one of the antiderivatives of f(x).The

MEAN

-

VALUE THEOREM

, however, shows that if F(x) is

any other antiderivative of f(x),then it must differ from

A(x) only by a constant: A(x) = F(x) + c. Noting that

A(a) = 0, we obtain: A(x) = F(x) – F(a).In particular,

the area under the curve y= f(x) from x= ato x= bis

A(b) = F(b) – F(a) . As this area is usually denoted

∫b

af(x)dx, this establishes the second part of the funda-

mental theorem of calculus:

If f(x) is a continuous function, and F(x) is any

antiderivative of f(x),then ∫b

af(x)dx = F(b) – F(a).

This second observation is the key result: it trans-

forms the very difficult problem of evaluating areas (via

limits of sums) into the much easier problem of finding

antiderivatives. For example, to compute the area

under the parabola y= x2from x= 3 to x= 12, simply