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If a wire with linear density $\rho(x, y, z)$ lies along a space curve $C,$ its moments of inertia about the $x-y-$ and $z$ -axes are defined as

$$I_{x}=\int_{C}\left(y^{2}+z^{2}\right) \rho(x, y, z) d s$$

$$I_{y}=\int_{C}\left(x^{2}+z^{2}\right) \rho(x, y, z) d s$$

$$I_{z}=\int_{C}\left(x^{2}+y^{2}\right) \rho(x, y, z) d s$$

Find the moments of inertia for the wire in Exercise 33.

$I_{x}=I_{y}=4 \sqrt{13} k \pi\left(1+6 \pi^{2}\right)$

$I_{z}=8 \sqrt{13} k \pi$

Vector Calculus

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