Finding the center of mass of a two dimensional object of constant density is a question that frequently occurs on my Mu Alpha Theta tests. I know of a long way to find it which I'll show in a moment using single integrals. I'm wondering if using double integrals can shorten this considerably.(adsbygoogle = window.adsbygoogle || []).push({});

Let f(x) be greater than or equal to g(x) on the domain [a,b]. The moments about the x and y axes are:

(1)[tex]M_{x}=p\int_{a}^{b}\left(\frac{f(x)+g(x)}{2}\right)[f(x)-g(x)]dx[/tex]

(2)[tex]M_{y}=p\int_{a}^{b}x[f(x)-g(x)]dx[/tex]

, where p is the planar lamina of uniform density.

The center of mass (x,y) is given by:

(3)[tex]x=\frac{M_y}{m}[/tex]

(4)[tex]y=\frac{M_x}{m}[/tex]

where [tex]m=p\int_{a}^{b}f(x)-g(x)dx[/tex]

Does anyone know how to do this same process with double integrals?! This is just way too much memorizing for me, and I say memorizing because I don't understand the derivation of it.

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# Double integrals and center of mass

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