Numerical Integration for multiple rectangles

[CodingNoob]

Hi, for my class I have to code a program which will find the integral of x^2 + 2 from 0 to 2 using the rectangle method for multiple rectangles n. I have to output n in one column and the corresponding integral in the other in a text file. In the code below I have loaded in my values of n from the n.txt file. This file is just a column containing 1 and 2. Therefore h will have 2 values. I then compute the area by letting the right hand side of the rectangle lie on the curve.

#include <iostream>
#include <fstream>
#include <iomanip>
#include <cmath>

using namespace std;

float f(float x);

int main(){


	int i;
	float area, height=0, h, n, a=0, b=2, x, y;
	
	ifstream inFile("n.txt");
	ofstream outFile("out.txt");
	

	outFile<<setw(7)<<"n";
	outFile<<setw(20)<<"Integral"<<endl;
	

	while(!inFile.eof()){
	inFile>>n;
	outFile<<setw(7)<<n;
		
	h=(b-a)/n;
	

	
	for(i=1; i<=n; i++){
	
	
 	height+=f(a+i*h);

	}
	area=height*h;
	
	outFile<<setw(20)<<area<<endl;
	}

	return 0;
	
}

float f(float x){
return pow(x,2)+2;
}

When n=1, I get the correct answer which is f(2)*h = 6*2 = 12. However, for n=2, I get an answer of 15 which shouldn't be the case if we consider two uniform rectangles: f(1)*h + f(2)*h with h=(2-0)/2 = 1. Therefore for 2 rectangles, the result should be (3*1)+(6*1)=9. I'm not sure how to correct my code. Could someone see what I'm doing wrong?

Thanks :)

[fg109]

I think your logic is wrong.

So n should be the number of rectangles you use to compute the integral right? The sum of the area of the rectangles would be the integral. Larger n means a more accurate answer.

Now how do you compute the area of a rectangle? You need to multiply its width and length, or base and height, whatever you want to call it. In this case, we know that the integral is from 0 to 2, represented by n rectangles. So width (or base) of each rectangle is 2.0 / n.

The length (or height) is computed from the calculation x^2 + 2. The x in this case would be i * width for the i-th rectangle. So to calculate the integral, you would need to compute the sum of width * (pow(i * width, 2) + 2) for n rectangles with i representing the current rectangle.

NOTE: I am assuming that the upper right corner of the rectangle touches the curve.

EDIT: Looking at your code, you actually have most of it right. You just need to fix line 35.

[CodingNoob]

Thanks for the reply. I'm not sure what I have to do to line 35. Wouldn't f(a+i*h) cover it because a=0? And you're right, having more rectangles should give me a more accurate answer, but my code computes the calculation incorrectly and I'm not sure why.

[fg109]

#include <iostream>
#include <string>
#include <sstream>
#include <iomanip>
#include <cmath>

using namespace std;

float f(float x);

int main()
{
    int numRects;
    float base, height, area, totalArea;
    const float RANGE_BEGIN = 0.0;
    const float RANGE_END = 2.0;
    const string STR = "1 2 3 4 5 6 7 8";
    istringstream in(STR);

    cout << setw(12) << "Rectangles" << setw(12) << "Area\n";
    cout << string(24, '-') << endl;

    while (in >> numRects)
    {
        totalArea = 0.0;
        base = (RANGE_END - RANGE_BEGIN) / numRects;
        for (int i = 1; i <= numRects; ++i)
        {
            height = f(i * base);
            area = base * height;
            totalArea += area;
        }
        cout << setw(12) << numRects;
        cout << setw(12) << setprecision(4) << totalArea << endl;
    }
    return 0;
}

float f(float x){
return pow(x,2)+2;
}

[CodingNoob]

Thanks for this! But how do I manipulate the code so the number of rectangles is loaded from a .txt file, and the table is outputted in a .txt file?

[fg109]

You showed that you could do file input/output in the code in your first post. Did someone else write that for you? If so, well then you need to at least do some work of your own.

[CodingNoob]

Nope, I wrote it myself and have been able to change it. Thanks for your help :)