Computers in Engineering - 308-208 Lecture 24
Computers in Engineering - 308-208

Lesson 24 - Learning Goals

24.1 Learn why Numerical Integration is useful

24.2 Learn several different Numerical Integration Techniques

24.3 Learn how to compare different Numerical Integration algorithms


This chapter is the first of the series that addresses some of the numerical techniques used. Numerical integration is very useful in situations where an exact solution to an integration problem cannot be found analytically.

The techniques described are the Midpoint, Trapezoidal and Simpson's rules. A comparison shows that Simpson's Rule is clearly superior to the others.

Associated with this chapter is also a graphical demonstration of the different methods. This is available on the associated diskette and is also accessible over the Internet.




We wish to integrate y=f(x) from x=a to x=b using n rectangular panels of equal width.

h = (b-a) /n
x0 = a
xi = xi-1 + h
Ai = h* f( (xi-1 + xi)/2 )
A Ai (i=1 i=n)
A h* [ f( (xi-1 + xi) /2 )] (i=1 i=n)

REAL FUNCTION MidPt(x0, x1, N)
! Estimate the area under F between x0 and x1
! using the Midpoint Rule with N panels.

   REAL x0, x1, x, h, Sum

   h = (x1 - x0) / N
   Sum = 0.0

   ! Add area of each panel, assuming unit width
   ! and multiplying by the true width h later.

   x = x0+h/2.0 ! Initial start is middle of first panel

   L1: DO k= 1,N
      Sum = Sum + F(x)
      x = x + h
   END DO L1

   MidPt = h*Sum


C Version

double midpoint(double x0, double x1, int n)
   double h, sum;

   /* Calculate panel width h; initialize Sum. */

   h= (x1 - x0) / n;
   x0= x0 + h/2;
   sum= 0;

/* Add area of each panel, multiplying
by the true width h only once later. */

   do {
      sum= sum + f(x0);
      x0= x0 + h;
   }while(x0 x1);
   return(sum * h);


Try taking average of the expressions for rectangular Ai, with f evaluated at left and right endpoints

Ai = [ h f(xi-1) + h f(xi) ]/2

Ai = h/2 [ f(xi-1) + f(xi) ]

Approximate Ai, as the area of trapezoid abcd.

Trapezoidal method appears to take 2 function evaluations per panel, but

Ai + Ai+1= h/2 [ f(xi-1) + f(xi) ] + h/2 [ f(xi) + f(xi+1) ]

Ai + Ai+1 = h/2 [ f(xi-1) + 2f(xi) + f(xi+1) ]

For n panels, need only n+1 evaluations of f.


We wish to integrate y=f(x) from x=a to x=b using n trapezoidal panels of equal width.

h= (b-a)/n

x0 = a

xn = b

xi = xi-1 + h

Ai = h/2 [ f(xi-1) + f(xi) ]

A Ai (i=1 i=n)

A h [ ( f(x0) + f(xn) ) / 2 + f(xi) ] (summing from i = 1 to n-1 )


In Trapezoidal rule, successive points (xi, yi) were joined with straight lines

Now, fit a parabola to 3 points (xi-2, yi-2), (xi-1, yi-1), and (xi, yi) for better accuracy

One panel (width 2h) now includes 3 points

h = (b-a)/n

b-a = nh

n must therefore be even

From geometry or Taylor series expansion, we find

Ai = h/3 [ f(xi-2) + 4f(xi-1) + f(xi) ] , for i = 2,4,6,…,n

Once more, combine expressions for adjacent panels and simplify.

Ai + Ai+2 = h/3 [ f(xi-2) + 4f(xi-1) + 2f(xi) + 4f(xi+1) + f(xi+2) ]

For n panels, again need only n+1 evaluations of f.


We wish to integrate y=f(x) from x=a to x=b using n/2 double parabolic panels of equal width.

h = (b-a)/n
x0= a
xn= b
xi= xi-1 + h
Ai = h/3 [ f(xi-2) + 4f(xi-1) + f(xi) ]
A Ai (i=1 i=n)

A h/3 [ f(x0) + f(xn) + 4 f(xi) + 2 f(xi) ]
    where first is from i=1 to n-1
    and second is from i=2 to n-2


- Arbitrary precision


/* more sophisticated adaptive integration method

using recursion */

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