MATH 251 Lecture 22

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Exam on Friday

Written Homework:

1. Calculating determinants
2. Polar gradient: $\mathrm{\nabla }f=f_x\hat{ı}+f_y\hat{ȷ}=f_r\hat{r}+\frac{f_{\theta }}{r}\hat{\theta }$
3. Classifying Quadratic Forms: $Q=a{x}^2+2bxy+c{y}^2\Rightarrow \left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ (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: $\mathrm{d}x\to \mathrm{d}r$, $\mathrm{d}y\to r\mathrm{d}\theta$

Example

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

$\underset{0}{\overset{1}{\int }}\underset{\sqrt{\dots }}{\overset{\sqrt{\dots }}{\int }}y\mathrm{d}x\mathrm{d}y$

In Polar Coordinates:

Formulas

$\begin{array}{rl}{C}_1:x^2+y^2-2y+1=1& \Rightarrow r=2\sin \theta \\ C_2:x^2+y^2=2& \Rightarrow r=\sqrt{2}\end{array}$

Region

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

$\begin{array}{r}2\sin \theta \le r\le \sqrt{2}0\le \theta \le \frac{\pi }{4}\end{array}$

$\underset{0}{\overset{\frac{\pi }{4}}{\int }}\underset{2\sin \theta }{\overset{\sqrt{2}}{\int }}{r}^2\sin \theta \mathrm{d}r\mathrm{d}\theta =\dots$

Mass / Center of Mass / Moment of Inertia

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

• The total mass in the region $R$ is $\underset{R}{\iint }\rho (x,y)\mathrm{d}A$
• The center of mass (not necessarily the middle of region) is $(\overline{x},\overline{y})=(M_x,M_y)$, where $\overline{x}$ is the mean value of $x$.
  • $\overline{x}=\frac{1}{\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s}}\underset{R}{\iint }x\cdot \rho (x,y)\mathrm{d}A$
  • $\overline{y}=\frac{1}{\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s}}\underset{R}{\iint }y\cdot \rho (x,y)\mathrm{d}A$

Example

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

Density = Constant = ${\rho }_0$

Mass = $\underset{R}{\iint }{\rho }_0\mathrm{d}A={\rho }_0\cdot \mathrm{A}\mathrm{r}\mathrm{e}\mathrm{a}=\frac{3\pi }{2}{\rho }_0$

Center of Mass:

• $\overline{x}=\underset{R}{\iint }x{\rho }_0\mathrm{d}A=0$
• $\overline{y}=\underset{R}{\iint }y{\rho }_0\mathrm{d}A=\frac{28}{9\pi }$