MATH 251 Lecture 25

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Volume

$V=\underset{R}{\iiint }\mathrm{d}V=\sum _{i}1(\mathrm{\Delta }{V}_i)$

Special Case

If $R$ is the region between the graphs of $f(x,y)$ and $g(x,y)$, then $R=\{(x,y,z)|f(x,y)\le z\le g(x,y)\wedge x,y\in S\subseteq {\mathbb{ℝ}}^3\}$ and

$V=\underset{S}{\iint }g(x,y)-f(x,y)\mathrm{d}A$

Area of Region

Area of region $R\in {\mathbb{ℝ}}^3$

$A=\underset{R}{\iint }\mathrm{d}A$

Example

Mass and center of mass of region bounded by $z=1-y^2$, $x+z=1$, $x=0$, and $z=0$. The region has a constant mass density of $\sigma (x,y,z)=7$

This is a wedge of a parabolic cylinder...

$M=\underset{-1}{\overset{1}{\int }}\underset{0}{\overset{1-y^2}{\int }}\underset{0}{\overset{1-z}{\int }}\mathrm{d}x\mathrm{d}y\mathrm{d}z$

Use this to find the center of mass. We already know by symmetry that $\overline{y}=0$

Cylindrical Coordinates

Polar Coordinates + $z$, so just replace $x$ and $y$ by $r$ and $\theta$.

Example

Integrate $f=z^2(x^2+y^2$ over the region inside the sphere $x^2+y^2+z^2=4$ and inside $(x-1{)}^2+y^2=1$

Convert to Cylindrical Coordinates:

• $f=r^2{z}^2$
• $r^2+z^2=4$
• $r=2\cos \theta$
• $\mathrm{d}V=r\mathrm{d}r\mathrm{d}\theta \mathrm{d}z$

Calculate ranges

• $z\in [-\sqrt{4-r^2},\sqrt{4-r^2}]$
• $r\in [0,2\cos \theta ]$
• $\theta \in [-\frac{\pi }{2},\frac{\pi }{2}]$

Evaluate Integral $\underset{-\frac{\pi }{2}}{\overset{\frac{\pi }{2}}{\int }}\underset{0}{\overset{2\cos \theta }{\int }}\underset{-\sqrt{4-r^2}}{\overset{\sqrt{4-r^2}}{\int }}{r}^3{z}^2\mathrm{d}z\mathrm{d}r\mathrm{d}\theta$