pi day: Calculate Pi in Azure Notebook

Interesting usage of iPython-like Notebooks for a web article:

This Pi Day, Calculate the Value of Pi for Yourself | WIRED
Calculating area of half-cicrle of radius r=1 as sum of vertical slices
The surface area of circle is r**2 * pi = pi  for r==1
Area of a slice is y * dx
x**2 + y**2 = r**2 = 1  =>  y = sqrt(1-x**2)

def calcPiSlices(n):
    dx = 2. / n # step size
    area = 0 # sum of all areas of half-circle
    for x in numpy.arange(-1, 1, dx):
        area = area + math.sqrt(1-x**2) * dx # area of a vertical slice
    return 2*area # *2 to get area of whole circle = pi

Here is the calculation in "real" Jupyter Notebook on 

  https://notebooks.azure.com/dragansr/libraries/calcPi

Pi - Wikipedia  |  5 Ways to Calculate Pi - wikiHow

faster: Nilakantha series method
π = 3 + 4/(2*3*4) - 4/(4*5*6) + 4/(6*7*8) - 4/(8*9*10) + 4/(10*11*12) - 4/(12*13*14) ... 

def calcPiNilakantha(n):
    pi = 3.0
    for x in range(2, n, 4):
        pi += (4 / (x*(x+1)*(x+2))) - (4 / ((x+2)*(x+3)*(x+4)))
    return pi