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 Skip Navigation LinksMath Help > Calculus > Calculus Theorems > Fundamental Th. of Calc > Def. of Integral

An Integral is, loosely speaking, the sum of an infinite number of infinitesimal numbers.  Mathematically speaking, you should consider a function f(x) where x ranges from a to b.  Now, divide that distance between a and b into n sections.  The integral from a to b of f(x) dx is the limit as n goes to infinity of the sum of n numbers, where each number is (1/n) times the value of the function in the ith section.

This relationship between the limit of a sum and the corresponding integral was described by Georg Friedrich Bernhard Riemann (1826-1866).  He was most famous for work in non-Euclidean geometry, differential equations, and number theory.  His results in physics and mathematics form the basis of Einstein's theory of general relativity.

Definition of Integral

Let f be defined on [a,b], and let D be a partition of [a,b] given by

a = x0 < x1 < x2 < ... < xn-1 < xn = b

where Dxi is the length of the ith subinterval.  If ci is any point in the ith sub-interval then the sum

n
S
i=1

f(ci) Dxi,   xi-1 <= ci <= xi

is called a Riemann sum of f for the partition D.

The length of the largest subinterval of a partition D is the norm of the partition, denoted IIDII.  If every subinterval is of equal length, the partition is regular and the norm is

IIDII = Dx = (b-a)/n

For a general partition, the norm is related to the number of subintervals of [a,b] in this way:

(b-a)/IIDII ≤ n

So as IIDII—>0, n —> ¥   (The converse is not true: if n—> ¥, it is not necessarily the case that IIDII—>0, because there could be one "big" partition and a whole bunch of infinitesimal ones, though the converse is true for regular partitions.)

Definite Integral

If f is defined on [a,b] and the limit of the Riemann sum

lim
IIDII—>0

n
S
i=i

f(ci) Dxi,   xi-1 <= ci <= xi

exists, then f is integrable on [a,b] and the limit is denoted by

ób
ô
õ
a

f(x) dx

How is this definition used?

The definition of the integral is used to prove the Fundamental Theorem of Calculus.

Indefinite Integral

The Fundamental Theorem of Calculus makes it clear that the definite integral

ób
ô
õ
a

f(x) dx

is a function of its limits only, and not a function of x.  In other words, x is just a parameter used to describe the integral, and not a variable that has meaning outside the integral.  Its use is akin to the use of x in the description of the set of even numbers

The set of even numbers is {x | x/2 is an integer}

In both the integral, above, and the set notation, immediately above, the symbol, x, is used as a parameter.  That is, it is used to relate parts of the definition to one another, but does not have meaning outside the definition.

From the Fundamental Theorem of Calculus, we know that if f(x) = F'(x) then the

ób
ô
õ
a

f(x) dx  =  F(b) - F(a)

If we think of a as a constant, F(a)=-C, and b as a variable, then

ób
ô
õ
a

f(x) dx  =  F(b) + C

No matter what the value of a is, there is some constant, C, that makes the statement true, so we need not write the lower limit.  Now the integral is a function of its upper limit only.  By convention, we re-use the symbol, x, as the upper limit of the integral (remember, the x inside the integral has no meaning outside the integral, so it doesn't conflict with this use) and we get to the indefinite integral, which we write as

ó
ô
õ

f(x) dx  =  F(x) + C

Thus the integral of f(x) is the antiderivative of f(x) plus a constant.

Graeme's comment: Boy is that confusing!  To learn this rigorously, you have to make very clear in your mind the meaning of the parameter, x, and how it differs from the use of x as an argument to the functions f(x) and F(x).  However, you'll be glad to know that this understanding isn't really important to the work of calculus, so once you learn it (if you ever do) you can just forget it (or at least, put it near the back of your mind).

Related pages in this website

Fundamental Theorem of Calculus -- if f is the derivative of F, then the integral from a to b of f(x)dx is F(b)-F(a)

Using the definite integral to calculate the centroid

Set Description Notation -- an example of the use of x as a "parameter"

 

The webmaster and author of this Math Help site is Graeme McRae.