Cartesian Form of Continuous Moment of Inertia

Theorem

Let $B$ be a rigid body which is rotating in space about some axis $\LL$.

Let $B$ be embedded in Cartesian $3$-space whose origin lies on $\LL$.

Let $I_{xx}$, $I_{yy}$, and $I_{zz}$ be the moments of inertia of $B$ about the coordinate axes $\mathbf O x$, $\mathbf O y$ and $\mathbf O z$ respectively.

Then:

$\ds I_{xx}$

$=$

$\ds \rho \int_B \paren {y^2 + z^2} \rd V$

$\ds I_{yy}$

$=$

$\ds \rho \int_B \paren {z^2 + x^2} \rd V$

$\ds I_{zz}$

$=$

$\ds \rho \int_B \paren {x^2 + y^2} \rd V$

where:

$\rho$ denotes the density of $B$

$\d V$ denotes a volume element of $B$.

Proof

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Sources

1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): moment of inertia
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): moment of inertia