Graphics Programs Reference
In-Depth Information
Trapezoidal Rule
E
f ( x )
Figure 6.2. Trapezoidal rule.
Area = I
h
x
x = a
x = b
1
2
If n
=
2 , wehave
=
( x
x 2 )
/
( x 1
x 2 )
=−
( x
b )
/
h . Therefore,
1
b
1
h
1
2 h ( b
h
2
a ) 2
A 1 =−
( x
b ) dx
=
=
a
Also
2 =
( x
x 1 )
/
( x 2
x 1 )
=
( x
a )
/
h ,sothat
b
1
h
1
2 h ( b
h
2
a ) 2
A 2 =
( x
a ) dx
=
=
a
SubstitutioninEq. (6.2a) yields
f ( b )] h
2
=
+
I
[ f ( a )
(6.3)
which isknownas the trapezoidal rule . It represents the area of the trapezoid inFig. 6.2.
The errorinthe trapezoidal rule
b
E
=
f ( x ) dx
I
a
is the area of the regionbetween f ( x ) and the straight-line interpolant, as indicated
in Fig. 6.2. It can beobtainedbyintegrating the interpolation errorinEq. (4.3):
b
) b
a
1
2!
1
2 f (
x 2 ) f (
E
=
( x
x 1 )( x
ξ
) dx
=
ξ
( x
a )( x
b ) dx
a
h 3
12 f (
1
12 ( b
a ) 3 f (
=−
ξ
)
=−
ξ
)
(6.4)
Composite Trapezoidal Rule
f ( x )
I
i
Figure 6.3. Composite trapezoidal rule.
h
x
x 1
x 2
x i
x i + 1
x n -1
x n
a
b

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