Information Technology Reference
In-Depth Information
by observing that the integral can be split into several parts. Let c
D
.a
C
b/=2,i.e.,
the midpoint between a and b.Thenwehave
Z
b
f.x/dx
D
Z
c
a
f.x/dx
C
Z
b
c
f.x/dx;
a
and thus, using (1.10) on each interval, we have
Z
b
f.x/dx
.c
a/
1
2
.f .a/
C
f.c//
C
.b
c/
1
2
.f .c/
C
f.b//
:
a
By observing that
1
2
.b
a/;
c
a
D
b
c
D
we can simplify this expression to obtain
Z
b
1
4
.b
a/ Œf .a/
C
2f .c/
C
f.b/:
f.x/dx
(1.11)
a
This method is illustrated in Fig.
1.3
.
Example 1.2.
Let us go back to the example above and use this slightly refined
method to compute an approximation of
Z
1:5
sin.x/ dx:
(1.12)
1
1.1
1.05
1
0.95
0.9
0.85
0.8
0.75
0.7
0.65
0.6
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
Fig. 1.3
The figure illustrates how the integral of f.x/
D
sin.x/ can be approximated by two
