# Simpsons Rule 10<-6 Error

I am making a Java function to calculate $\int_{0}^{\pi/2}x/sin(x) dx$ using Simpsons rule. When the function equals $0$ or $0/\sin(0)$ should I just add $1$ (limit) or $10^{-10}$ (limit). My professor said to use $10^{-10}$ why would this be?

public static double simpsonsRuleFunction1(double valueN, double valueA, double valueB, double valueDx) {

double e = 0.0;
double simpsonsRule = 0.0;
double valueHolder = 0.0;

valueN = 2;
valueA = 0;
valueB = (Math.PI)/2;

for(int i = 1; i<=valueN+1 ; i++){
valueDx = (valueB-valueA)/valueN;
e = valueA + ((i-1)*valueDx);

if (i==1) {
// Limit as x -> 0
simpsonsRule += Math.pow(10,-10);

}
else if ((i % 2 == 0) && ( i > 1) && (i < valueN+1 ))  {
simpsonsRule += 4*(e/((Math.sin(e))));
}
else if ((i % 2 != 0) && ( i > 1) && (i < valueN+1 )) {
simpsonsRule += 2*(e/((Math.sin(e))));
}
else if (i == valueN+1 ) {
simpsonsRule += (e/((Math.sin(e))));

}

}
simpsonsRule = simpsonsRule *((valueDx)/3);

System.out.println("\nsimpsonsRule" + simpsonsRule);

while(Math.abs(valueHolder - simpsonsRule) > Math.pow(10,-6)) {
System.out.println("\nValueHolder" + valueHolder);
valueHolder = simpsonsRule;
valueN +=2;
valueDx = (valueB-valueA)/valueN;
simpsonsRule = 0;
for(int i = 1; i<=valueN + 1; i++){
e = valueA + ((i-1)*valueDx);

if (i==1) {
// Limit as x -> 0
simpsonsRule += Math.pow(10,-10);

}
else if (i % 2 == 0) {
simpsonsRule += 4*(e/((Math.sin(e))));
}
else if ((i % 2 != 0) && ( i > 1) && (i < valueN + 1)) {
simpsonsRule += 2*(e/((Math.sin(e))));
}
else if (i == valueN + 1) {
simpsonsRule += (e/((Math.sin(e))));

}

}
simpsonsRule = simpsonsRule *((valueDx)/3);

}
return valueN;
}

• Perhaps your professor meant to use the value obtained by substituting $x=10^{-10}$ ? – Paul Nov 3 '15 at 15:53

You can simplify your code by removing redundant checks in if statements. You can also pull out the valueDx which is constant in the loop:

double e = 0.0;
double simpsonsRule = 0.0;
double valueHolder = 0.0;

valueN = 2;
valueA = 0;
valueB = (Math.PI)/2;

valueDx = (valueB-valueA)/valueN;
simpsonsRule = 1.0 + Math.PI/2;
for(int i = 2; i<=valueN ; i++){
e = valueA + ((i-1)*valueDx);

if (i % 2 == 0)  {  //even
simpsonsRule += 4*(e/((Math.sin(e))));
}
else { //odd
simpsonsRule += 2*(e/((Math.sin(e))));
}
}

simpsonsRule = simpsonsRule *((valueDx)/3);


Also it looks like you repeat this same loop twice in your code for some reason. I havn't run this so there may still be a few bugs especially in the indices etc but this is homework so I wont do it all for you =). You can look at a python or c variant which does what you want on Wikipedia

The python implementation is particularly close to yours. To answer your question though, you are evaluating between 0 and $\pi/2$. Thus the line:

simpsonsRule = 1.0 + (Math.PI)/2


corresponds to adding $f(0) + f(\pi/2)$ where $f(x)=\frac{x}{\sin(x)}$. In the limit $f(0)->1$ while $f(\pi/2)=\pi/2$.

• Oh so I would use the limit of f(0) -> 1 When I == 1 instead of 10^-10? – Julius A. Nov 3 '15 at 3:13
• Yes because f(0)->1. Is it possible your professor misunderstood you when you showed him the code? Also don't forget to go over the indicies since I have not run the above code to see if it works. – James Nov 3 '15 at 3:23