FIRST YEAR CALCULUS
W W L CHEN
c
W W L Chen, 1987, 2005.
This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990.
It is available free to all individuals, on the understanding that it is not to be used for financial gain,
and may be downloaded and/or photocopied, with or without permission from the author.
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Chapter 9
NUMERICAL INTEGRATION
9.1. Introduction
In Chapters 7 and 8, we have discussed some analytic techniques for evaluating integrals. However,
many integrals that arise in science and engineering resist attack by even the most sophisticated analytic
techniques. In such instances, we may have to accept a rather poor and perhaps even not entirely
satisfactory second best, and attempt to make reasonable approximations by numerical techniques.
9.2. The Trapezium Rule
Suppose that we wish to evaluate an integral
B
A
f
(
x
)d
x,
where the function
f
(
x
) is finite and continuous in the closed interval [
A, B
]. If we draw the curve
y
=
f
(
x
), then the value of the integral is the same as the area bounded by the curve
y
=
f
(
x
) and the
lines
y
= 0,
x
=
A
and
x
=
B
(the reader should draw a diagram).
A first, and rather crude, approximation to the integral is to take the area of the trapezium with
vertices at the points (
A,
0), (
B,
0), (
A, f
(
A
)) and (
B, f
(
B
)). In other words, we take the approximation
(1)
B
A
f
(
x
)d
x
≈
1
2
(
B
−
A
)(
f
(
A
) +
f
(
B
))
.
In practice, however, we take more points than just
A
and
B
. Consider the dissection
A
=
x
0
< x
1
< . . . < x
n
=
B
Chapter 9 : Numerical Integration
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