8/9/2019 Iib-10.Numerical Integration 80-83
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NUMERICAL IN TEGRATION
S y n o p s i s :
1. Trapezoidal Rule : Let y = f(x) be a continuous function
defined on [a, b]. Divide the interval on
x-axis into n equal parts each of length h =n
ab . Let a = x 0, x 1, x 2, , x n =b be the abscissae of
the successive points of division and y r = f(x r) for r = 0, 1,
2, , n. Then = dx)x(fb
a
= ++++ ...)x(f2)x(f2)x(f[2h
210 2f(x n1 ) + f(x n)]
= )}]x(f...)x(f)x(f{2)x(f)x(f[2h
1n21n0 +++++
= )]y...yy(2)yy[(2h
1n21n0 +++++
2. Simpsons Rule : Let y = f(x) be a continuous function defined
on [a, b]. Divide the interval on x-
axis into n = 2m (even integer) equal parts each of length h
=m2
abn
ab = . Let a = x 0, x 1, x 2, ,x n
= x 2m = b be the abscissae of the successive points of division
and y r = f(x r) for r = 0, 1, 2n. Then
...)x(f4)x(f2)x(f4)x(f[3h
dx)x(f 3210
b
a
++++= )]x(f)x(f4 m21m2 ++
= )}x(f...)x(f)x(f{4)}x(f)x(f[{3h
1m231m20 +++++
+ )}]x(f...)x(f)x(f{2 2m242 +++
= )]y...yy(4)yy[(3h 1m231m20 +++++ +
)]y...yy(2 2m242 +++
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