This document discusses numerical integration techniques including the trapezoidal rule, Simpson's 1/3 rule, Simpson's 3/8 rule, and Gaussian integration formulas. It provides the formulas for calculating integration numerically using these methods and notes that accuracy increases with smaller interval widths h. Errors are estimated to be order h^2 for trapezoidal rule and order h^4 for Simpson's rules.

1. Numerical Methods - Numerical
Integration
N. B. Vyas
Department of Mathematics,
Atmiya Institute of Tech. and Science, Rajkot (Guj.)
niravbvyas@gmail.com
N. B. Vyas Numerical Methods - Numerical Integration

2. Numerical Integration
Let I =
b
a
y dx where y = f(x) takes the values y0, y1, . . . , yn for
x0, x1, . . . , xn
N. B. Vyas Numerical Methods - Numerical Integration

3. Numerical Integration
Let I =
b
a
y dx where y = f(x) takes the values y0, y1, . . . , yn for
x0, x1, . . . , xn
Let us divide the interval (a, b) into n sub-intervals of width h so
that x0 = a, x1 = a + h = x0 + h, x2 = x0 + 2h, . . .,
xn = x0 + nh = b then
N. B. Vyas Numerical Methods - Numerical Integration

4. Numerical Integration
Let I =
b
a
y dx where y = f(x) takes the values y0, y1, . . . , yn for
x0, x1, . . . , xn
Let us divide the interval (a, b) into n sub-intervals of width h so
that x0 = a, x1 = a + h = x0 + h, x2 = x0 + 2h, . . .,
xn = x0 + nh = b then
I =
b
a
y dx =
x0+nh
x0
f(x) dx
N. B. Vyas Numerical Methods - Numerical Integration

5. Numerical Integration
Let I =
b
a
y dx where y = f(x) takes the values y0, y1, . . . , yn for
x0, x1, . . . , xn
Let us divide the interval (a, b) into n sub-intervals of width h so
that x0 = a, x1 = a + h = x0 + h, x2 = x0 + 2h, . . .,
xn = x0 + nh = b then
I =
b
a
y dx =
x0+nh
x0
f(x) dx
Trapezoidal rule:
b=x0+nh
a=x0
f(x)dx =
h
2
[(y0 + yn) + 2 (y1 + y2 + .... + yn)]; h =
b − a
n
If the number of strips is increased; that is, h is decreased, then
the accuracy of the approximation is increased.
N. B. Vyas Numerical Methods - Numerical Integration

6. Numerical Integration
Simpson’s
1
3
rd rule:
N. B. Vyas Numerical Methods - Numerical Integration

7. Numerical Integration
Simpson’s
1
3
rd rule:
x0+nh
x0
f(x)dx = h
3 [(y0 + yn) + 4(y1 + y3 + ....)
+2(y3 + y4 + ....)]; h = b−a
n
N. B. Vyas Numerical Methods - Numerical Integration

8. Numerical Integration
Simpson’s
1
3
rd rule:
x0+nh
x0
f(x)dx = h
3 [(y0 + yn) + 4(y1 + y3 + ....)
+2(y3 + y4 + ....)]; h = b−a
n
while applying this rule, the given interval must be divided into
even number of equal sub-intervals. i.e. n must be even.
N. B. Vyas Numerical Methods - Numerical Integration

9. Numerical Integration
Simpson’s
1
3
rd rule:
x0+nh
x0
f(x)dx = h
3 [(y0 + yn) + 4(y1 + y3 + ....)
+2(y3 + y4 + ....)]; h = b−a
n
while applying this rule, the given interval must be divided into
even number of equal sub-intervals. i.e. n must be even.
Simpson’s
3
8
th rule:
N. B. Vyas Numerical Methods - Numerical Integration

10. Numerical Integration
Simpson’s
1
3
rd rule:
x0+nh
x0
f(x)dx = h
3 [(y0 + yn) + 4(y1 + y3 + ....)
+2(y3 + y4 + ....)]; h = b−a
n
while applying this rule, the given interval must be divided into
even number of equal sub-intervals. i.e. n must be even.
Simpson’s
3
8
th rule:
x0+nh
x0
f(x)dx = 3h
8 [(y0 + yn) + 3(y1 + y2 + y4 + y5 + ....)
+2(y3 + y6 + ....)]; h = b−a
n
N. B. Vyas Numerical Methods - Numerical Integration

11. Numerical Integration
Simpson’s
1
3
rd rule:
x0+nh
x0
f(x)dx = h
3 [(y0 + yn) + 4(y1 + y3 + ....)
+2(y3 + y4 + ....)]; h = b−a
n
while applying this rule, the given interval must be divided into
even number of equal sub-intervals. i.e. n must be even.
Simpson’s
3
8
th rule:
x0+nh
x0
f(x)dx = 3h
8 [(y0 + yn) + 3(y1 + y2 + y4 + y5 + ....)
+2(y3 + y6 + ....)]; h = b−a
n
while applying this rule, the number of sub-intervals should be
taken as a multiple of 3 i.e. n must be multiple of 3
N. B. Vyas Numerical Methods - Numerical Integration

12. Numerical Integration
Gaussian Integration Formula:
1
−1
f(t)dt =
n
i=1
wif(ti)
It should be noted here that, t = ±1 is obtained by setting
x =
1
2
[(b + a) + t (b − a)]
N. B. Vyas Numerical Methods - Numerical Integration

13. Numerical Integration
Gaussian Integration Formula: The following table gives the
values for n = 2, 3, 4, 5
N. B. Vyas Numerical Methods - Numerical Integration

14. Example
Ex. Evaluate
1
0
e−x2
dx by using Gaussion integration formula for
n = 3.
Sol. Here, we have to first convert the given integral from 0 to 1 into
an integral from −1 to 1. x = 1
2 [(b + a) + t (b − a)], a = 0 and
b = 1
∴ x =
t + 1
2
⇒ dx =
dt
2
∴
1
0
exp(−x2)dx =
1
2
1
−1
exp −
1
4
(t + 1)2 dt
N. B. Vyas Numerical Methods - Numerical Integration

15. Error
Error in Quadrature Formula:
If yp is a polynomial representing the function y = f(x) in the
interval [x0, xn] the error in the quadrature formula is given by
E =
xn
x0
f(x) =
xn
x0
ypdx
N. B. Vyas Numerical Methods - Numerical Integration

16. Error
Error in Trapezoidal rule:
|error| ≤ (b − a)
h2
12
|f (M)|
where f (M) = max |f 0(x)|, |f 1(x)|, ..., |f n−1(x)|
∴ error is of order h2
total error =
dh3
12
y 0 + y 1 + ... + y n−1
N. B. Vyas Numerical Methods - Numerical Integration

17. Error
Error in Simpson’s
1
3
rd rule:
|error| ≤ (b − a)
h4
180
|f4
(M)|
where f4(M) = max |y4
0|, |y4
2|, ..., |y4
n−2|
∴ error is of order h4
total error =
h5
90
y4
0 + y4
2 + ... + y4
n−2
N. B. Vyas Numerical Methods - Numerical Integration

18. Error
Error in Simpson’s
3
8
th rule:
|error| ≤ (b − a)
h4
80
|f4
(M)|
where f4(M) = max |y4
0|, |y4
3|, ..., |y4
n−3|
∴ error is of order h4
total error =
3h5
80
y4
0 + y4
3 + ... + y4
n−3
N. B. Vyas Numerical Methods - Numerical Integration