MATH 251 Lecture 23

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Applications of Integrals

Given mass density $\rho (x,y)$ of ideal laminar region (perfectly flat)

mass

$\mathrm {mass} =\iint _{R}\rho (x,y)\,\mathrm {d} A$

center of mass

${\bar {x}}={\frac {1}{\mathrm {mass} }}\iint _{R}x\rho (x,y)\,\mathrm {d} A$

${\bar {y}}={\frac {1}{\mathrm {mass} }}\iint _{R}y\rho (x,y)\,\mathrm {d} A$

moment of inertia

first moment about $y$ axis: $\iint x\rho (x,y)\,\mathrm {d} A$

first moment about $x$ axis: $\iint y\rho (x,y)\,\mathrm {d} A$

second moment about $y$ axis: $\iint x^{2}\rho (x,y)\,\mathrm {d} A$

second moment about $x$ axis $\iint y^{2}\rho (x,y)\,\mathrm {d} A$

third moment about $y$ axis: $\iint x^{3}\rho (x,y)\,\mathrm {d} A$

third moment about $x$ axis: $\iint y^{3}\rho (x,y)\,\mathrm {d} A$

etc.

Example

Find the 2nd moments of:

• $\rho (x,y)=1+x+y$
• $y=1-{\frac {x}{3}}$
• $y={\frac {x}{3}}-1$
• $x\in \left[0,3\right]$

Second moment about $y$ axis $\int _{0}^{3}\int _{{\frac {x}{3}}-1}^{1-{\frac {x}{3}}}x^{2}(1+x+y)\,\mathrm {d} y\,\mathrm {d} x=45-{\frac {162}{5}}$

Second moment about $x$ axis $\int _{0}^{3}\int _{{\frac {x}{3}}-1}^{1-{\frac {x}{3}}}y^{2}(1+x+y)\,\mathrm {d} y\,\mathrm {d} x=\ldots$

Gaussian Curve

Find $I=\int _{-\infty }^{\infty }\mathrm {e} ^{-{\frac {x^{2}}{2}}}\,\mathrm {d} x$

Instead, we're going to find $I^{2}=I\cdot I$, which can be simplified to

$\iint _{\mathbb {R} ^{2}}\mathrm {e} ^{-{\frac {x^{2}+y^{2}}{2}}}\,\mathrm {d} A$

Converting this to polar coordinates gives

$\int _{0}^{2\pi }\int _{0}^{\infty }r\mathrm {e} ^{-{\frac {r^{2}}{2}}}\,\mathrm {d} r\,\mathrm {d} \theta$

Which evaluates to $2\pi$ by u-substitution. Therefore, $I={\sqrt {2\pi }}$, so the normalized probabylity density function (pdf) $N(0,1)$ is given by

$p(x)={\frac {1}{\sqrt {2\pi }}}\mathrm {e} ^{-{\frac {x^{2}}{2}}}$