Center of Mass/Examples/Uniform Lamina

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Theorem

Let $\LL$ be a uniform lamina embedded in a cartesian plane in the shape of the area between the curve $\map f x$, the straight lines $x = a$ and $x = b$, and the $x$-axis.

Let the area of $\LL$ be $A$.


Then the coordinates $\tuple {\bar x, \bar y}$ of the center of mass of $B$ are given by:

\(\ds A \bar x\) \(=\) \(\ds \int_a^b x y \rd x\)
\(\ds A \bar y\) \(=\) \(\ds \dfrac 1 2 \int_a^b y^2 \rd y\)


Proof


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