## How to calculate the moment of inertia of the rigid body?

I don't know how to calculate the following rigid bodies with different geometries, can anybody help me?

Thin spherical shell: I=(2/3)MR^2

Solid sphere: I=(2/5)MR^2

 Recognitions: Homework Help Science Advisor Use the definition of moment of inertia: $$I = \int r^2 dm$$ In the case of the shell the element of mass is $dm = M {dA} /{4 \pi R^2}$ where $dA = R^2 \sin \theta d\theta d\phi[/tex]. The distance to a point on the shell from the z-axis is [itex]R^2 \sin^2 \theta$ so $$I = \frac {M}{4 \pi R^2} \int_{0}^{2 \pi} d\phi \int_{-\pi /2}^{\pi /2}R^4 \sin^3 \theta d\theta$$ from which the desired result follows. In the case of the solid sphere you will work with a volume integral.