Integral-Approximation

Using the Monte Carlo method.

Algorithm steps:

1. Generate independently and uniformly n points from a rectangle [a, b] x [0, M]
   for a given M >= sup{ f(x) : x ∈ [a,b] }
   
2. Count a number of generated points that lay under the curve of a function f(x)
   A point (x, y) lies under the curve of f(x) if y <= f(x) ( = C )
   
3. The approximation is C/n * ( b−a ) * M, where ( b−a ) * M is an area of the rectangle
   

Sample Result:

• f(x) = x^1/3

• a = 0
  b = 8
  M = 3
  
  Which gives: [0, 8] x [0, 3]
  
  

• Blue points - repeats of the algorithm for a given n

• Red points - an average value of repeats for a given n
  
  
  

• The actual value of such integral of f(x) is 12!