implementing riemann sum for the area of a circle - c++

Question

The following code is designed to implement this formula:
void integrate ( double R , double E ) //radius and acceptable error, entered by user
{
int i = 1; //iterator
int n = 1; //number of rectangles
double x = 0;
double width = 0; //stores width of each rectangle
double area = 0; //stores calculated area
double error = E + 1; //stores difference between calculated area and pi*(radius)^2
while ( error > E ) //loop continues as long as error is greater than the acceptable limit entered by user
{
width = R / n; //sets the width of rectangles
area = 0; //with this line, calculated area converges to zero. without it, calculated area increases without bound
while ( i <= n )
{
x = i; //assign iterator value to a separate variable to avoid data type issues in calculation
area += width * sqrt ( pow ( R , 2.0 ) - pow ( width * ( x - 0.5 ) , 2.0 ) ); //riemann integral formula
i++;
}
n++; //increase number of rectangles by 1 to be used in next loop
area *= 4; //forumla uses one quarter of a circle, multiply by 4 to get total area
error = abs ( area - ( pi * pow ( R , 2.0 ) ) ); //error calculation
cout << " \n Calculated area = " << area << "\n" //debugging output
<< " \n Error = " << error << "\n";
cin.get();
}
cout << "\n Area: " << area << "\n" //output to user
<< "\n Error: " << error << "\n"
<< "\n Iterations: " << i << "\n";
return;
}
As the comment says, when the line "area = 0;" is included in the first loop, the area converges to zero, and when it is removed, the area increases without bound. I've been working on this one function for the better part of a week without progress. The rest of the program is completed, tested, and works fine. I have to assume that there's either an error in the logic or the math that I just can't see. Any help would be appreciated.
I hope that the comments are sufficient. I'll provided additional clarification as necessary.
(This is a self-contained function. No other part of the program affects it or is affected by it aside from the user input.)