#include <iostream>
#include <cmath>
using namespace std;

double calculateArea( double leftLimit, double rightLimit, double tolerance, double area );
double lHeight( double leftLimit );
double rHeight( double rightLimit );


int main()
{
    double leftLimit, rightLimit, tolerance, initialArea, area1;

    cout << "This is a demonstartion of integrating x^3 + 2." << endl;
    cout << "Enter the left & right limits of integration: ";
    cin >> leftLimit >> rightLimit;
    cout << endl;

    cout << "Enter the desired error tolerance: ";
    cin >> tolerance;
    cout << endl;

    //function is f(x) = x^3 + 2
    initialArea = (rightLimit-leftLimit)*(( lHeight( leftLimit ) +  rHeight( rightLimit ))/2 );
    area1 = calculateArea( leftLimit, rightLimit, tolerance, initialArea );
    cout << "The area is approximately " << area1 << endl;

    return 0;
}

double calculateArea( double leftLimit, double rightLimit, double tolerance, double area )
{
    double mid = (leftLimit + rightLimit)/2;
    double area2 = (mid-leftLimit)*(( lHeight( leftLimit ) +  rHeight( mid ))/2 ) +
        (rightLimit-mid)*(( lHeight( mid ) + rHeight( rightLimit ))/2 );
    if( fabs(area-area2) < tolerance )
        return area2;
    else
    {
        return calculateArea( leftLimit, mid, tolerance/2, area2 ) + calculateArea( mid, rightLimit, tolerance/2, area2 );
    }
}

double lHeight( double leftLimit )
{
    return pow( leftLimit, 3 ) + 2;
}

double rHeight( double rightLimit )
{
    return pow( rightLimit, 3 ) + 2;
}