Geometry Approximation of an Integral I am looking for help for two things:
1.how precise this approximation is (such as Big O or percent error)
2. How efficient the algorithm for time / space usage.
I created an algorithm to estimate an integral. The algorithm creates squares and adds the estimate of the remain sum times the area together. Since this function is 2d, the area is multiplied by f(x, y) at a point. The function in this program is f(x, y) = x + (a * y).
The documented code is below.


def f(x:float, y:float, a:float)-&gt;float:
    &quot;&quot;&quot;
    param x:float, x within the function domain 
    param y:float, y within the function domain 
    param a:float, a paramter of the curve 
    Output: Number of function at the point 
    &quot;&quot;&quot;
    return x + (a * y)
    
class Square:
    def __init__(self, xi:float, yi:float, xf:float, yf:float)-&gt;None:
        &quot;&quot;&quot;
        param xi:float, smaller coordinate of x 
        param yi:float, smaller coordinate of y 
        param xf:float, larger cooordinate of x 
        &quot;&quot;&quot; 
        self.xi = xi 
        self.yi = yi 
        self.xf = xf 
        self.yf = yf 
        
    def estimate(self, a:float)-&gt;float:
        &quot;&quot;&quot;
        param a:float, a from the curve paramter 
        Output: a term to add to estimate the intergal 
        &quot;&quot;&quot;
        xd = self.xf - self.xi 
        yd = self.yf - self.yi 
        if xd &lt; 0:
            xd = xd * -1 
        if yd &lt; 0:
            yd = yd * -1 
        area_delta = xd * yd 
        est = f(self.xi, self.yi, a) 
        return area_delta * est  
        

def main()-&gt;None:
    &quot;&quot;&quot;&quot;
    Input: None 
    Output:None 
    Note: driver program 
    &quot;&quot;&quot;
    x = [1.0, 2.0, 3.0]
    y = [2.0, 3.0, 4.0]
    a = 1.0 
    add = 0.0 
    square = None  
    for row in range(len(x)):
        for col in range(len(x)):
            if row &lt;= col:
                continue 
            square = Square(x[row], y[row], x[col], y[col])
            add = add + square.estimate(a)
    print(add)

if __name__ == &quot;__main__&quot;:
    main() 
    
``
You are just re-implementing the abs built-in here. (If a number is negative make it positive):
    xd = self.xf - self.xi 
    yd = self.yf - self.yi 
    if xd &lt; 0:
        xd = xd * -1 
    if yd &lt; 0:
        yd = yd * -1 

In general I do not understand why you have a whole class defined. I think this should be written more compactly because the actual calculations performed are very simple.
To be more user friendly I would have a function having as input the domain of integration and the function to integrate like:
def weighted_area(xrange, yrange, function, precision):

An argument should be the precision in case the user wants to change the integration step.