Numerical Methods for Eng
[ENGR 391]
[Lyes KADEM 2007]
CHAPTER VI
Numerical Integration
Topics

Riemann sums

Trapezoidal rule

Simpson’s rule

Richardson’s extrapolation

Gauss quadrature
rule
Mathematically, integration is just finding the area under a curve from one point to another. It is
represented by
, where the symbol
is an integral sign, the numbers
a
and
b
are the
lower and upper limits of integration, respectively, the function
f
is the integrand of the integral, and
x
is the variable of integration. Figure 1 represents a graphical demonstration of the concept.
∫
b
a
dx
x
f
)
(
∫
Why are we interested in integration: because most equations in physics are differential equations
that must be integrated to find the solution(s). Furthermore, some physical quantities can be obtained
by integration (example: displacement from velocity).
The problem is that sometimes integrating analytically some functions can easily become laborious.
For this reason, a wide variety of numerical methods have been developed to find the integral.
Figure.6.1
 Integration.
I. Riemann Sums
Let
f
be defined on the closed interval [
a
,
b
], and let
∆
be an arbitrary partition of [
a
,
b
] such as:
a
=
x
0
<
x
1
<
x
2
< … <
x
n1
<
x
n
=
b
, where
∆
x
i
is the length of the
i
th
subinterval.
If
c
i
is any point in the
i
th
subinterval, then the sum
∑
=
−
≤
≤
Δ
n
i
i
i
i
i
i
x
c
x
x
c
f
1
1
,
)
(
Numerical Integration
81
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[ENGR 391]
[Lyes KADEM 2007]
∆
x
i
x
1
x
2
...
x
i1
x
i
…
x
n1
x
n
is called a Riemann Sum of the function
f
for the partition
∆
on the interval [
a
,
b
].
For a given partition
∆
, the length of the longest subinterval is called the norm of the partition. It is
denoted by

∆

(the norm of
∆
). The following limit is used to define the definite integral:
∑
=
→
Δ
=
Δ
n
i
i
i
I
x
c
f
1
0
)
(
lim
This limit exists if and only if for any positive number
ε
, there exists a positive number
δ
such that for
every partition
∆
of [
a
,
b
] with

∆

<
δ
, it follows that
ε
<
Δ
−
∑
=
n
i
i
i
x
c
f
I
1
)
(
for any choice of the numbers
c
i
in the
i
th
subinterval of
∆
.
If the limit of a Riemann Sum of
f
exists, then the function
f
is said to be integrable over [
a
,
b
] and that
the Riemann Sums of
f
on [
a
,
b
] approach the number
I
.
∑
=
→
Δ
=
Δ
n
i
i
i
I
x
c
f
1
0
)
(
lim
,
Where
∫
=
b
a
dx
x
f
I
)
(
Example
Find the area of the region between the parabola
y = x
2
and the xaxis on the interval [0,
4.5]. Use Riemann’s Sum with four partitions.
2. TRAPEZOIDAL RULE
Trapezoidal rule is based on the NewtonCotes formula that if we approximate the integrand by an n
th
order polynomial, then the integral of the function is approximated by the integral of that n
th
order
polynomial.
0
,
1
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 Spring '08
 KRACZEK
 Lyes KADEM

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