a = var("a") # The lower bound
b = var("b") # The upper bound
c = var("c") # This is the width of each rectangle

f(x) = 2*x**2 + 3*x + 1  # The function we are finding the area under, between the bounds a and b.

def Riemann(f, a, b, c):
    """
    A function to find the area under another function between the paramiters a and b.
"""
    d = 0
    while b > a:
        d += (f(b) * (1/c)) # summing the areas of the rectangle (with f(b) denoting the hight and 1/c the width)
        b -= (1/c)
    return d




plot_data =  [Riemann(f, 1, 5, c) for c in srange(1,10,1)] # Giving intiger values of c between one and ten, (so the width of the rectangles decreas).
p = list_plot(plot_data)
p.show()