Numerical Integration with Unspecified Parameter in C++ - numerical-integration

Question

I have written a simple code to calculate an integral numerically with both the midpoint and trapezoidal rule. It works fine, but now I would like to modify it so that it can calculate the same integral but with an unspecified parameter. So instead of the integral of e^(-x^2), I would like to calculate the integral of e^(-\alpha*x^2). However, I am stuck here. I tried to just declare a new double alpha without initializing it, but that clearly takes me no where. Is there a way to do such a thing in C++? Thank you in advance.
//=============================================================
// A simple program to numerically integrate a Gaussian over
// the interval a to b
//=============================================================
#include <iostream>
#include <cmath>
using namespace std;
// function implementing the midpoint rule
double midpoint(double a, double b, int n){
double sum = 0.0;
double width = (b-a)/n;
for (int i = 1; i < n; i++){
sum += exp((-1)*(a+(i+0.5)*width)*(a+(i+0.5)*width))*width;
}
return sum;
}
//function implementing the trapezoidal rule
double trapezoidal(double a, double b, int n){
double width = (b-a)/n;
double sum = (width/2)*(exp(-a*a)+exp(-b*b));
for (int i = 1; i < n; i++){
sum += 2*exp((-1)*(a+i*width)*(a+i*width))*width/2;
}
return sum;
}
int main(){
cout << "THIS IS A SIMPLE PROGRAM TO INTEGRATE A GAUSSIAN OVER A CERTAIN INTERVAL." << endl;
int n;
double a, b;
cout << "Enter the lower and upper limit of integration as doubles" << endl;
cin >> a >> b;
cout << "Enter the number of partitions" << endl;
cin >> n;
double actual = 0.886207;
double midRule = midpoint(a,b,n);
double trapRule = trapezoidal(a,b,n);
cout << "For the function e^(-x^2) integrated from "<< a << " to " << b << endl;
cout << "The analytic value of the integral is: " << actual << endl;
cout << "The midpoint value of the integral for " << n << " partitions is: " << midRule << endl;
cout << "The trapezoidal value of the integral for " << n << " partitions is: " << trapRule << endl;
cout << "The percent error for the midpoint is: " << (abs(actual-midRule)/actual)*100 << "%" << endl;
cout << "The percent error for the trapezoidal is: " << (abs(actual-trapRule)/actual)*100 << "%" << endl;
return 0;
}