import numpy as np

# This function provides the midpoint quadrature rule.
def Midpoint( a, b, N, f ) :
    # [a,b] : inteval
    # N : number of subintervals
    # f : fonction to integrate using the midpoint rule

    # size of the subintervals
    H = (b - a) / N
 
    # quadrature nodes
    x = np.linspace(a+H/2, b-H/2, N)

    # approximate integral 
    return H * np.sum( f(x) );
   
    
# This function provides the trapezoidal quadrature rule.
def Trapezoidal( a, b, N, f ) :
    # [a,b] : inteval
    # N : number of subintervals
    # f : fonction to integrate using the trapezoidal rule

    # size of the subintervals
    H = (b - a) / N

    # quadrature nodes
    x = np.linspace(a, b, N+1)

    # approximate integral 
    return H/2 * ( f(a) + f(b) ) + H * sum( f(x[1:N]) );

# This function just provides the Simpson quadrature rule.
def Simpson( a, b, N, f ) :
    # [a,b] : inteval
    # N : number of subintervals
    # f : fonction to integrate using the Simpson rule

    # size of the subintervals
    H = (b - a) / N
    # points defining intervals
    x = np.linspace(a, b, N+1)
    
    # left quadrature nodes
    xl = x[0:-1]    
    # right quadrature nodes
    xr = x[1:]
    # middle quadrature nodes
    xm = (xl+xr)/2

    # approximate integral 
    return H/6 * sum( f(xl) + 4*f(xm) + f(xr) );