Center of Mass - Unbalanced Ring

Calculus Level 5

A massive circular ring of unit radius is centered on the origin within the \(xy\) plane. The ring parameters and mass density are given below:

\[x = \cos(\theta) \hspace{1cm} y = \sin(\theta) \hspace{1cm} 0 \leq \theta \leq \ 2 \pi \hspace{1cm} \frac{dm}{d\theta} = e^{-\theta}\]

How far away from the origin is the ring's center of mass (to three decimal places)?

Note: The constant \(e\) is Euler's Number


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