MATH 251 Lecture 22

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Exam on Friday

Written Homework:

1. Calculating determinants
2. Polar gradient: $\nabla f=f_{x}{\hat {\imath }}+f_{y}{\hat {\jmath }}=f_{r}{\hat {r}}+{\frac {f_{\theta }}{r}}{\hat {\theta }}$
3. Classifying Quadratic Forms: $Q=ax^{2}+2bxy+cy^{2}\Rightarrow {\begin{pmatrix}a&b\\c&d\end{pmatrix}}$ (positive, negative, mixed, degenerate)

Integrals in Polar Coordinates

• convert region to polar coordinates: $R_{x,y}$ to $R_{r,\theta }$
• convert function to polar coordinates: $x=r\cos {\theta }$, $y=r\sin {\theta }$
• convert differentials to polar coordinates: $\mathrm {d} x\rightarrow \mathrm {d} r$, $\mathrm {d} y\rightarrow r\mathrm {d} \theta$

Example

Region in first quadrant of $x^{2}+y^{2}=2$ excluding the part bounded by $x^{2}+(y-1)^{2}=1$

$\int _{0}^{1}\int _{\sqrt {\dots }}^{\sqrt {\dots }}y\,\mathrm {d} x\,\mathrm {d} y$

In Polar Coordinates:

Formulas

${\begin{aligned}C_{1}:~x^{2}+y^{2}-2y+1=1&~\Rightarrow ~r=2\sin {\theta }\\C_{2}:~x^{2}+y^{2}=2&~\Rightarrow ~r={\sqrt {2}}\end{aligned}}$

Region

${\begin{aligned}0\leq \theta \leq \arcsin {\left({\frac {r}{2}}\right)}\\0\leq r\leq {\sqrt {2}}\end{aligned}}$

${\begin{aligned}2\sin {\theta }\leq r\leq {\sqrt {2}}0\leq \theta \leq {\frac {\pi }{4}}\end{aligned}}$

$\int _{0}^{\frac {\pi }{4}}\int _{2\sin {\theta }}^{\sqrt {2}}r^{2}\sin {\theta }\,\mathrm {d} r\,\mathrm {d} \theta =\ldots$

Mass / Center of Mass / Moment of Inertia

Mass density function $\rho (x,y)$, meaning that for a small area $\left(x^{*},y^{*}\right)$, the mass of that region is $\rho (x^{*},y^{*})\cdot A$.

• The total mass in the region $R$ is $\iint _{R}\rho (x,y)\mathrm {d} A$
• The center of mass (not necessarily the middle of region) is $({\bar {x}},{\bar {y}})=(M_{x},M_{y})$, where ${\bar {x}}$ is the mean value of $x$.
  • ${\bar {x}}={\frac {1}{\mathrm {mass} }}\iint _{R}x\cdot \rho (x,y)\,\mathrm {d} A$
  • ${\bar {y}}={\frac {1}{\mathrm {mass} }}\iint _{R}y\cdot \rho (x,y)\,\mathrm {d} A$

Example

Use region as half-washer with outer radius 2, inner radius 1, and θ between 0 and π

Density = Constant = $\rho _{0}$

Mass = $\iint _{R}\rho _{0}\,\mathrm {d} A=\rho _{0}\cdot \mathrm {Area} ={\frac {3\pi }{2}}\rho _{0}$

Center of Mass:

• ${\bar {x}}=\iint _{R}x\rho _{0}\,\mathrm {d} A=0$
• ${\bar {y}}=\iint _{R}y\rho _{0}\,\mathrm {d} A={\frac {28}{9\pi }}$