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Calculate the moment of inertia of a solid cone about an axis through its centre. The cone has a mass M and height h, and the radius of its circular base is R.
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Using integration - disc by disc.

I = Sum Ix = Sum Mx*y^2/2 = Sum pAxdx*y^2/2 = Sum p(pi*y^2)dx*y^2/2
I = p(pi/2) Sum y^4 dx

As dx -->0,
I = p(pi/2) Integral y^4 dx, x from 0 to h, y = (R/h)x
I = p(pi/2) Integral (R/h)^4 x^4 dx
I = p(pi/2)(R/h)^4(h^5/5)
I = p(pi/2)(R^4)(h/5)
I = (pi/10)(R^4)*h*p

p = M/V, V = pi*R^2*h/3
p = 3M/(pi*R^2*h)

I = (3/10)MR^2
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