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#include <iostream>
#include <cmath>
 
using namespace std;
 
double rect_integral(double a, double b, int n, int choice);
double trap_integral(double a, double b, int n, int choice);
 
class FunctionInterface
{
public:
    virtual ~FunctionInterface() {};
    virtual double Calculate(double x) = 0;
    virtual std::string GetName() = 0;
};
 
class Function1: public FunctionInterface
{
public:
    FunctionInterface() {}
 
    double Calculate (double x)
    {
        return 2.0*pow(x,5.0) +pow(x,3.0) -10*x +2;
    }
 
    std::string GetName()
    {
        return m_Name;
    }
private:
    static const std::string m_Name = "2x^5 + x^3 -10x+2";
};
 
class Function2: public FunctionInterface
{
public:
    FunctionInterface() {}
 
    double Calculate (double x)
    {
        return 6*pow(x,2.0) -x +10;
    }
 
    std::string GetName()
    {
        return m_Name;
    }
private:
    static const m_Name = "6x^2 - x +10";
};
 
 
 
int main () {
    FunctionInterface* functions[2];
    functions[0] = new Function1();
    functions[1] = new Function2();
 
  start:
    int choice;
    int n;
    double a;
    double b;
    bool trap = false;
    bool rect = false;
 
    cout << "Functions available: \n\n";
    for (int i=0; i<2; i++)
        cout << "  " << i << "  " << functions[i].GetName() << '\n';
    cout << endl;
 
    cout << "Choose a function (1, 2, 3, 4, 5, other(quit)) " << endl;
    cin >> choice;
 
    if (choice >= 0 && choice < 2)
    {
        char method = ' ';
        do {
            cout << "Would you like to calculate the area using the rectangle,"
                 << " trapezoid, or both (1, 2, 3): " << endl;
            cin >> method;
        } while (method < '1' || method > '3');
 
        switch (method) {
            case '1':
                rect = true;
                cout <<"How many rectangles do you want? " << endl;
                break;
            case '2':
                trap = true;
                cout <<"How many trapezoids do you want? " << endl;
                break;
            case '3':
                trap = true;
                rect = true;
                cout <<"How many rectangles/trapezoids do you want? " << endl;
        }
 
        cin >> n;
 
        cout <<"Please select a starting point, a: " << endl;
        cin >> a;
 
        cout <<"Please select an ending point, b: " << endl;
        cin >> b;
 
        if (rect) {
            cout << "The area under " << function[choice]->GetName()
                 << " between " << a << " and " <<  b
                 << " is: " <<  rect_integral(a, b, n, *function[choice]) <<" in rectangles" << endl;
        }
 
        if (trap) {
            cout << "The area under " << function[choice]->GetName()
                 << " between " << a << " and " <<  b
                 << " is: " <<  trap_integral(a, b, n, *function[choice]) << " in trapezoids" << endl;
        }
    }
int again;
cout << "Would you like to go again? Yes = 1 " <<endl;
cin >> again;
 
if (again == 1)
   goto start;
 
    return 0;
}
// end of int main() start of functions
 
 
double rect_integral(double a, double b, int n, FunctionInterface& function)
{
    double width = (b-a)/n;
 
    double x = a;
    double total_area = 0;
    for(int i = 1; i <= n; i++) {
        x = width *i;
        total_area += width * function.Calculate(x);
    }
 
    return total_area;
}   
 
double trap_integral(double a, double b, int n, FunctionInterface& function)
{
    double x =a; 
 
    double interiml, interimr;
    double total_area = 0;
    for(int i = 1; i <= n; i++) {
        x = width *i;
        interiml = width * fx(x);
        x = width*i;
        interimr = width * function.Calculate(x);
        total_area += (interiml + interimr)/2;
    }
 
    return total_area;
}
 

prog.cpp:20:23: error: ISO C++ forbids declaration of ‘FunctionInterface’ with no type [-fpermissive]
prog.cpp:32:39: error: invalid in-class initialization of static data member of non-integral type ‘const string {aka const std::basic_string<char>}’
prog.cpp:32:39: error: initializer invalid for static member with constructor
prog.cpp:32:39: error: (an out of class initialization is required)
prog.cpp:32:39: error: ‘Function1::m_Name’ cannot be initialized by a non-constant expression when being declared
prog.cpp: In member function ‘int Function1::FunctionInterface()’:
prog.cpp:20:26: warning: no return statement in function returning non-void [-Wreturn-type]
prog.cpp: At global scope:
prog.cpp:38:23: error: ISO C++ forbids declaration of ‘FunctionInterface’ with no type [-fpermissive]
prog.cpp:50:18: error: ‘m_Name’ does not name a type
prog.cpp: In member function ‘int Function2::FunctionInterface()’:
prog.cpp:38:26: warning: no return statement in function returning non-void [-Wreturn-type]
prog.cpp: In member function ‘virtual std::string Function2::GetName()’:
prog.cpp:47:16: error: ‘m_Name’ was not declared in this scope
prog.cpp: In function ‘int main()’:
prog.cpp:70:51: error: request for member ‘GetName’ in ‘functions[i]’, which is of pointer type ‘FunctionInterface*’ (maybe you meant to use ‘->’ ?)
prog.cpp:109:42: error: ‘function’ was not declared in this scope
prog.cpp:115:42: error: ‘function’ was not declared in this scope
prog.cpp: In function ‘double trap_integral(double, double, int, FunctionInterface&)’:
prog.cpp:153:13: error: ‘width’ was not declared in this scope
prog.cpp:154:32: error: ‘fx’ was not declared in this scope
prog.cpp: In member function ‘virtual std::string Function2::GetName()’:
prog.cpp:48:5: warning: control reaches end of non-void function [-Wreturn-type]

Standard output is empty