Definite Integral

In Section 5.2 we investigated the limit of a finite sum for a function defined over a closed
interval [a, b] using n subintervals of equal width (or length), In this section
we consider the limit of more general Riemann sums as the norm of the partitions of [a, b]
approaches zero. For general Riemann sums the subintervals of the partitions need not
have equal widths. The limiting process then leads to the definition of the definite integral
of a function over a closed interval [a, b].
Limits of Riemann Sums
The definition of the definite integral is based on the idea that for certain functions, as the
norm of the partitions of [a, b] approaches zero, the values of the corresponding Riemann
sb - ad>n.
5.3
sums approach a limiting value I. What we mean by this converging idea is that a Riemann
sum will be close to the number I provided that the norm of its partition is sufficiently
small (so that all of its subintervals have thin enough widths). We introduce the symbol
as a small positive number that specifies how close to I the Riemann sum must be, and the
symbol as a second small positive number that specifies how small the norm of a partition
must be in order for that to happen. Here is a precise formulation.
d
P
344 Chapter 5: Integration
DEFINITION The Definite Integral as a Limit of Riemann Sums
Let ƒ(x) be a function defined on a closed interval [a, b]. We say that a number I
is the definite integral of ƒ over [a, b] and that I is the limit of the Riemann
sums if the following condition is satisfied:
Given any number there is a corresponding number such that
for every partition of [a, b] with and any choice of
in we have
` a
n
k=1
ƒsckd ¢xk - I ` 6 P.
ck [xk-1, xk] ,
P = 5x0 , x1, ? , xn6 7P7 6 d
P 7 0 d 7 0
gnk
=1ƒsckd ¢xk
Leibniz introduced a notation for the definite integral that captures its construction as
a limit of Riemann sums. He envisioned the finite sums becoming an infinite
sum of function values ƒ(x) multiplied by “infinitesimal” subinterval widths dx. The
sum symbol is replaced in the limit by the integral symbol whose origin is in the
letter “S.” The function values are replaced by a continuous selection of function values
ƒ(x). The subinterval widths become the differential dx. It is as if we are summing
all products of the form as x goes from a to b. While this notation captures the
process of constructing an integral, it is Riemann’s definition that gives a precise meaning
to the definite integral.
Notation and Existence of the Definite Integral
The symbol for the number I in the definition of the definite integral is
which is read as “the integral from a to b of ƒ of x dee x” or sometimes as “the integral
from a to b of ƒ of x with respect to x.” The component parts in the integral symbol also
have names:
?
?
?
?
???????
The function is the integrand.
x is the variable of integration.
When you find the value
of the integral, you have
evaluated the integral.
Upper limit of integration
Integral sign
Lower limit of integration
Integral of f from a to b
a
b
f (x) dx
L
b
a
ƒsxd dx