MATH 251 Lecture 20

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Integration

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

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 $(\overline{x},\overline{y})$ will be the weighted averages value of $x$ and $y$:

$\begin{array}{rl}\overline{x}& =\frac{1}{M}\underset{R}{\iint }x\rho (x,y)\mathrm{d}A\\ \overline{y}& =\frac{1}{M}\underset{R}{\iint }y\rho (x,y)\mathrm{d}A\\ M& =\underset{R}{\iint }\rho (x,y)\mathrm{d}A\end{array}$

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