## Product of Inertia

The product of inertia of a body with density function
$\rho (x,y,z)$
$xy$
plane is
$I_{xy} = \int_z \int_y \int_x xy \rho (x,y,z) dx dy dz$

$xz$
plane is
$I_{xz} = \int_z \int_y \int_x xz \rho (x,y,z) dx dy dz$

$yz$
plane is
$I_{yz} = \int_z \int_y \int_x yz \rho (x,y,z) dx dy dz$

Example: If
$\rho (x,y,z) =x+yz$
over the region bounded by
$0 \leq x \leq 1 , 0 \leq y \leq 2 , 1 \leq z \leq 3$
then
\begin{aligned} I_{xy} &= \int^3_1 \int^2_0 \int^1_0 xy (x+yz) dx dy dz \\ &= \int^3_1 \int^2_0 \int^1_0 x^2y +xy^2 z dx dy dz \\ &= \int^3_1 \int^2_0 [ \frac{x^3y}{3} + \frac{x^2y^2z}{2}]^1_0 dy dz \\ &= \int^3_1 \int^2_0 \frac{y}{3} + \frac{y^2z}{2} dy dz \\ &= \int^3_1 [\frac{y^2}{6} + \frac{y^3 z}{6} ]^2_0 dz \\ &= \int^3_1 \frac{2}{3} + \frac{4z}{3} dz \\ &= [\frac{2z}{3} + \frac{2z^2}{3}]^3_1 \\ &= (\frac{2 \times 3}{3} + \frac{2 \times 3^2}{3} ) - (\frac{2 \times 1}{3} + \frac{2 \times 1^2 }{3}) = \frac{20}{3} \end{aligned}

The product of inertia has a physical meaning.
$I_xy = I_yx$
is the inertia of a mass rotating around the
$x$
axis against its rotation around the
$y$
axis.