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The CenterOfMass command in the Student MultivariateCalculus package is the primary tool for computing a centroid or a center of mass. This command will return the inert integrals defining $\stackrel{\&conjugate0;}{x}$ and $\stackrel{\&conjugate0;}{y}$, or the value of these integrals, or, for a lamina, a graph displaying the region $R$, the surface $\mathrm{\ρ}$, and the point $\left(\stackrel{\&conjugate0;}{x}\,\stackrel{\&conjugate0;}{y}\right)$.
There are also two relevant task templates that implement the CenterOfMass command, one each for a double integral in Cartesian or polar coordinates. However, these task templates iterate in a fixed order: $\mathrm{dy}\mathrm{dx}$ for the Cartesian case; and $\mathrm{dr}d\mathrm{theta;}$ for the polar.
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Of course, the MultiInt command in the Student MultivariateCalculus package (or the modified int command in the Student VectorCalculus package) can be used to integrate over triangles, or over all or part of disks and ellipses. For each such region, a robust task template implements the underlying command, but the MultiInt command can be more easily implemented from the Context Panel when the Student MultivariateCalculus package has been installed.
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