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       #Post#: 7--------------------------------------------------
       Numerical Integration II
   DIR By: mathnathan
       Date: July 11, 2013, 8:30 am
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       Here is the program we wrote in class together on Wednesday...
       It is also attached to this post for you to download.
       --- Code ---
       
       #include <stdio.h>
       #include <cmath>
       
       double lhrIntegrate( int n, float a, float b );
       double f( float x );
       
       int main() {
       
       float a = 2;
       float b = 6;
       double val;
       double answer = 32.0/3.0;
       double minErr = 1000;
       double err;
       int optimalN;
       double optimalVal;
       
       for( int n = 0; n < 5000; n++ ) {
       if( n%100 == 0 ) {
       printf( "On iteration %d\n", n );
       }
       val = lhrIntegrate( n, a, b );
       err = std::abs(answer-val);
       //printf( "val = %f\n", val );
       //printf( "answer = %f\n", answer );
       //printf("err = %f\n", err);
       if( err < minErr ) {
       minErr = err;
       optimalN = n;
       optimalVal = val;
       }
       }
       
       printf( "integral value = %f\n", val );
       printf( "minimum err = %f\n", minErr );
       printf( "Optimal n = %d\n", optimalN );
       printf( "Optimal val = %f\n", optimalVal );
       
       return 0;
       
       }
       
       double f( float x ) {
       return 4.0-(x-4.0)*(x-4.0);
       }
       
       double lhrIntegrate( int n, float a, float b ) {
       
       //printf( "n = %d\n", n );
       //printf( "a = %f\n", a );
       //printf( "b = %f\n", b );
       double sum = 0;
       float step = (b-a)/n;
       float xi = a;
       
       for( int i = 0; i < n; i++ ) {
       //printf("Iteration %d\n", i );
       //printf("\txi = %f\n", xi);
       //printf("\tarea = %f\n", step*f(xi));
       sum += step*f(xi);
       //printf("\ttotal sum = %f\n", sum );
       xi += step;
       }
       
       return sum;
       
       }
       
       --- End Code ---
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