272 integral calculus

Approximating length and area

∫

An even better approximation could be made using

more points and consequently shorter line segments.

The actual length of the curve would be the

LIMIT

value

of these improved approximations as we use shorter

and shorter line segments connecting more and more

points on the curve (the sum of infinitely short line seg-

ments). Similarly, the area under a curve drawn in the

plane can be approximated as the sum of the areas of a

finite number of rectangles drawn under the curve.

Using narrower and narrower rectangles will give bet-

ter and better approximations. The actual area under

the curve is the limit value of these approximations (the

sum of infinitely narrow rectangles).

Any process that involves segmenting a quantity

into manageable pieces, summing, and taking the limit

of these sums as the process is refined falls under the

category of integral calculus. Traditionally, integral

calculus is first taught as the process of finding the

area under a curve y= f(x) over an interval [a,b]. The

area denoted

∫b

af(x)dx

is called a definite integral, and is defined to be the

limit, as htends to zero, of the sums of the areas of

rectangles of width at most h, used to approximate the

area of the curve as described above. Such an approxi-

mation with rectangles is called a Riemann sum, in

honor of the German mathematician B

ERNHARD

R

IE

-

MANN

(1826–66), whose work led mathematicians to

show that this approach is indeed mathematically

sound, in particular that, under reasonable conditions,

all ways of approximating the area under the curve

lead to the same limit value.

G

OTTFRIED

W

ILHELM

L

EIBNIZ

(1646–1716), one of

the inventors of

CALCULUS

, introduced the symbol ∫to

represent an integral. He thought of it as an elongated

S denoting sum, and he called the theory of integration

calculus summatorius. Swiss mathematician Johann

Bernoulli (1667–1748) of the famous B

ERNOULLI FAM

-

ILY

worked with Leibniz in developing the theory, but

he preferred the name calculus integralis and the use of

a capital letter I as the sign of integration. (In Latin, the

word integralis means “making up a whole.”) The two

gentlemen settled on a happy compromise of using

Bernoulli’s name for the theory and Leibniz’s symbol

for the integral.

The idea of using a limit to calculate the areas of

curved figures, or the lengths of curved paths, has been

used by scholars from the time of A

RCHIMEDES OF

S

YRACUSE

in the third century

B

.

C

.

E

. to the time of

P

IERRE DE

F

ERMAT

in the middle of the 17th-century. In

practice, however, the techniques employed to perform

these calculations have always been extremely difficult

and complicated. The great achievement of Leibniz and

I

SAAC

N

EWTON

(1642–1727), independent discoverers

of calculus, was to recognize that integration is simply a

process of reverse differentiation, today called

ANTIDIF

-

FERENTIATION

. This result, known as the

FUNDAMENTAL

THEOREM OF CALCULUS

, essentially states that to find

the area under a curve y= f(x) over an interval [a,b],

look for a function F(x) whose

DERIVATIVE

is f(x).Then:

∫b

af (x)dx = F(b) – F(a)

The function F(x) is called an antiderivative of f. The

right-hand side of this equation is often abbreviated

as F(x)

|

b

a. Thus, for example, the area under the

parabola y= x2from x= 0 to x= 2, is given by

. This remarkable

result obviates all need to work with complicated

limits.

To highlight the interplay between integration and

reverse differentiation, the antiderivative of a function

f(x) is usually denoted ∫f(x)dx and is called the indefi-

nite integral of f. It is defined up to a

CONSTANT OF

INTEGRATION

. Thus ∫f(x)dx is a function whose deriva-

tive is f(x) (whereas the definite integral ∫b

af(x)dx is a

number equal to the area under the curve y= f(x) over

the interval [a,b)]). The thrust of integral calculus is the

development of methods for finding the antiderivatives

of functions.

Since the derivative of the sum of two functions is

the sum of the derivatives, and the derivative of a