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Problem 38 Hard Difficulty

If a wire with linear density $ \rho(x, y, z) $ lies along a space curve $ C $, its $ \textbf{moments of inertia} $ about the $ x $-, $ y $-, and $ z $-axes are defined as
$$ I_x = \int_C (y^2 + z^2) \rho(x, y, z) ds $$
$$ I_y = \int_C (x^2 + z^2) \rho(x, y, z) ds $$
$$ I_z = \int_C (x^2 + y^2) \rho(x, y, z) ds $$

Find the moments of inertia for the wire in Exercise 35.

Answer

$$
\begin{array}{c}{I_{x}=I_{y}=4 \sqrt{13} k \pi\left(1+6 \pi^{2}\right)} \\ {I_{z}=8 \sqrt{13} k \pi}\end{array}
$$

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Video Transcript

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