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- Jan 26, 2012

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**Background Info**: The tendency of a lamina to resist a change in rotational motion about an axis is measured by its

*moment of inertia*about that axis. If a lamina occupies a region $R$ of the $xy$-plane, and if its density function $\delta(x,y)$ is continuous on $R$, then the moments of inertia about the $x$-axis, $y$-axis, and $z$-axis are denoted by $I_x$, $I_y$, and $I_z$ respectively, and are defined by

\[I_x= \iint\limits_R y^2\delta(x,y)\,dA,\qquad I_y = \iint\limits_R x^2\delta(x,y)\,dA\]

\[I_z= \iint\limits_R (x^2+y^2)\delta(x,y)\,dA\]

**Problem**: Consider the circular lamina that occupies the region described by the inequalities $0\leq x^2+y^2\leq a^2$. Assuming that the lamina has constant density $\delta$, show that

\[I_x=I_y=\frac{\delta\pi a^4}{4},\qquad I_z=\frac{\delta\pi a^4}{2}.\]

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