Cylindrical - Maple Help

pAverage Value of a Function in Cylindrical Coordinates

Description

In the cylindrical coordinate system, where the point $\left(x,y,z\right)$ has coordinates $\left(r,\mathrm{θ},z\right)$, and

 (1)

$\left(r,\mathrm{θ}\right)$ are the polar coordinates of $\left(x,y\right)$, determine the average value of a function.

Average Value of a Function in Cylindrical Coordinates

Integrand

 > ${z}$
 ${z}$ (2)

Region: $\left\{{z}_{1}\left(r,\mathrm{θ}\right)\le z\le {z}_{2}\left(r,\mathrm{θ}\right),{r}_{1}\left(\mathrm{θ}\right)\le r\le {r}_{2}\left(\mathrm{θ}\right),a\le \mathrm{θ}\le b\right\}$

${z}_{1}\left(r,\mathrm{θ}\right)$

 > ${r}$
 ${r}$ (3)

${z}_{2}\left(r,\mathrm{θ}\right)$

 > ${1}$
 ${1}$ (4)

${r}_{1}\left(\mathrm{\theta }\right)$

 > ${0}$
 ${0}$ (5)

${r}_{2}\left(\mathrm{θ}\right)$

 > ${1}$
 ${1}$ (6)

$a$

 > ${0}$
 ${0}$ (7)

$b$

 >
 ${2}{}{\mathrm{π}}$ (8)

Inert Integral:

(Note automatic insertion of Jacobian.)

 > $\mathrm{Student}\left[\mathrm{MultivariateCalculus}\right]\left[\mathrm{FunctionAverage}\right]\left(,{z}=..,{r}=..,{\mathrm{θ}}=..,\mathrm{coordinates}=\mathrm{cylindrical}\left[{r}{,}{\mathrm{θ}}{,}{z}\right],\mathrm{output}=\mathrm{integral}\right)$
 $\frac{{{∫}}_{{0}}^{{2}{}{\mathrm{π}}}{{∫}}_{{0}}^{{1}}{{∫}}_{{r}}^{{1}}{z}{}{r}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{z}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{r}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{θ}}}{{{∫}}_{{0}}^{{2}{}{\mathrm{π}}}{{∫}}_{{0}}^{{1}}{{∫}}_{{r}}^{{1}}{r}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{z}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{r}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{θ}}}$ (9)

Value

 > $\mathrm{Student}\left[\mathrm{MultivariateCalculus}\right]\left[\mathrm{FunctionAverage}\right]\left(,{z}=..,{r}=..,{\mathrm{θ}}=..,\mathrm{coordinates}=\mathrm{cylindrical}\left[{r}{,}{\mathrm{θ}}{,}{z}\right]\right)$
 $\frac{{3}}{{4}}$ (10)

 Commands Used