Proving moment of inertia of hollow sphere with y^2+x^2=R^2 method

[parsa418]

Could anyone prove the moment of inertia of a thin walled hollow sphere using the y^2 + x^2 = r^2. I have only studied up to single variable calculus. I can take regular integrals but not multivariable integrals. I don't want to use the angle method or any polar coordinate systems except in the part where I might use it for trigonometric substitution to evaluate the integral.
Thanks

 

[Andrew Mason]

Could anyone prove the moment of inertia of a thin walled hollow sphere using the y^2 + x^2 = r^2. I have only studied up to single variable calculus. I can take regular integrals but not multivariable integrals. I don't want to use the angle method or any polar coordinate systems except in the part where I might use it for trigonometric substitution to evaluate the integral.
Thanks

You are missing a z^2 there for the third dimension.

The moment of inertia is simply:

I = \int r^2dm

Let dm be a slice of the sphere of thickness dy. Can you express dm in terms of area of that ring and density? Then integrate from y = -R to +R

AM