Chapter 14.8, Problem 49ES

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 $\delta \left(x,y\right)$ is continuous on R, then the moments of inertia about the x-axis, the y-axis, and the z-axis are denoted by $I_x{I}_{y,}\text{and}{I}_z,$ respectively, and are defined by

$I_x={\displaystyle \underset{R}{\iint }{y}^2\delta \left(x,y\right)dA,}{I}_y={\displaystyle \underset{R}{\iint }{y}^2\delta \left(x,y\right)dA,I_z={\displaystyle \underset{R}{\iint }\left(x^2+y^2\right)\delta \left(x,y\right)dA,}}$

Use these definitions in Exercise 49 and 50.

Consider the rectangular lamina that occupies the region described by the inequalities $0\le x\le a,\text{and}0\le y\le b.$ Assuming that the lamina has constant density $\delta$ show that

$I_x=\frac{\delta a{b}^3}{3},I_y=\frac{\delta a^{3}b}{3},I_z=\frac{\delta ab\left(a^2+b^2\right)}{3}$