Moment of Inertia-Ellipse

By implicitly differentiating the equation for the ellipse, we get the following relationship between the infinitesimals:

$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \\ dy = dx \, \frac{- b^2 x}{a^2 y}$

The rest is fairly straightforward. Integrate over one quadrant and multiply by four.

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import math

a = 3.0
b = 2.0

lam = 1.0

dx = 10.0**(-7.0)

#############################

I = 0.0

x = 0.0

while x <= a:

    right = 1.0 - (x/a)**2.0
    y = math.sqrt(4.0*right)

    dy = dx * (-b*b*x)/(a*a*y)

    dL = math.hypot(dx,dy)
    dm = lam * dL

    dI = dm*(x**2.0 + y**2.0)

    I = I + dI

    x = x + dx

I = 4.0*I

#############################

print dx
print I

#>>> 
#1e-05
#99.082438713
#>>> ================================ RESTART ================================
#>>> 
#1e-06
#99.1751859155
#>>>
#>>> 
#1e-07
#99.2858145478
#>>>