MATH 251 Lecture 25

« previous | Wednesday, March 28, 2012 | next »

Volume

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

Special Case

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

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

Area of Region

Area of region $R\in \mathbb {R} ^{3}$

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

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...

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

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

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

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

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