MATH 251 Lecture 20

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Integration

$\iint _{R}f(x,y)\,\mathrm {d} A$

Applications:

1. Volume under a surface
2. Mass of a body given a mass density equation $\rho (x,y)$

Center of Mass of a body

For a body $R$ with a mass density $\rho (x,y)$, the center of mass $({\bar {x}},{\bar {y}})$ will be the weighted averages value of $x$ and $y$:

${\begin{aligned}{\bar {x}}&={\frac {1}{M}}\iint _{R}x\rho (x,y)\,\mathrm {d} A\\{\bar {y}}&={\frac {1}{M}}\iint _{R}y\rho (x,y)\,\mathrm {d} A\\M&=\iint _{R}\rho (x,y)\,\mathrm {d} A\end{aligned}}$

Where $M$ represents the total mass of the object.