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

$$ I_{x}=\int_{C} y^{2} \rho(x, y) d s \quad I_{y}=\int_{C} x^{2} \rho(x, y) d s $$

$I_{x}=k\left(\frac{\pi}{2}-\frac{4}{3}\right)$

$I_{y}=k\left(\frac{\pi}{2}-\frac{2}{3}\right)$

Vector Calculus

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Missouri State University

Oregon State University

Harvey Mudd College

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