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

$ I_x = \int_C y^2 \rho(x, y) ds $ $ I_y = \int_C x^2 \rho(x, y) ds $

Find the moments of inertia for the wire in Example 3.

$\boldsymbol{I}_{x}=k\left(\frac{1}{2} \pi-\frac{4}{3}\right), \boldsymbol{I}_{y}=k\left(\frac{1}{2} \pi-\frac{2}{3}\right)$

Vector Calculus

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