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Average Value of a Function in Spherical Coordinates





 Description In the spherical coordinate system, where the point $\left(x,y,z\right)$ is defined by $\left(\mathrm{ρ},\mathrm{φ},\mathrm{θ}\right)$ and related by , , and , find the average value of a function.

Average Value of a Function in Spherical Coordinates

($\mathrm{φ}$ = colatitude, measured down from $z$-axis)

Integrand

 > ${\mathrm{ρ}}$
 ${\mathrm{ρ}}$ (1)

Region: $\left\{{\mathrm{ρ}}_{1}\left(\mathrm{φ},\mathrm{θ}\right)\le \mathrm{ρ}\le {\mathrm{\rho }}_{2}\left(\mathrm{φ},\mathrm{θ}\right),{\mathrm{φ}}_{1}\left(\mathrm{θ}\right)\mathit{\le }\mathrm{φ}\mathit{\le }{\mathrm{φ}}_{2}\left(\mathrm{θ}\right),a\le \mathrm{θ}\le b\right\}$

${\mathrm{ρ}}_{1}\left(\mathrm{φ},\mathrm{θ}\right)$

 > ${0}$
 ${0}$ (2)

${\mathrm{\rho }}_{2}\left(\mathrm{φ},\mathrm{θ}\right)$

 > ${1}$
 ${1}$ (3)

${\mathrm{φ}}_{1}\left(\mathrm{θ}\right)$

 > ${0}$
 ${0}$ (4)

${\mathrm{φ}}_{2}\left(\mathrm{θ}\right)$

 > $\frac{{\mathrm{π}}}{{6}}$
 $\frac{{1}}{{6}}{}{\mathrm{π}}$ (5)

$a$

 > ${0}$
 ${0}$ (6)

$b$

 >
 ${2}{}{\mathrm{π}}$ (7)

Inert Integral:

 > $\mathrm{Student}\left[\mathrm{MultivariateCalculus}\right]\left[\mathrm{FunctionAverage}\right]\left(,{\mathrm{ρ}}=..,{\mathrm{φ}}=..,{\mathrm{θ}}=..,\mathrm{coordinates}=\mathrm{spherical}\left[{\mathrm{ρ}}{,}{\mathrm{φ}}{,}{\mathrm{\theta }}\right],\mathrm{output}=\mathrm{integral}\right)$
 $\frac{{{∫}}_{{0}}^{{2}{}{\mathrm{π}}}{{∫}}_{{0}}^{\frac{{1}}{{6}}{}{\mathrm{π}}}{{∫}}_{{0}}^{{1}}{{\mathrm{ρ}}}^{{3}}{}{\mathrm{sin}}{}\left({\mathrm{φ}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{ρ}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{φ}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{θ}}}{{{∫}}_{{0}}^{{2}{}{\mathrm{π}}}{{∫}}_{{0}}^{\frac{{1}}{{6}}{}{\mathrm{π}}}{{∫}}_{{0}}^{{1}}{{\mathrm{ρ}}}^{{2}}{}{\mathrm{sin}}{}\left({\mathrm{φ}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{ρ}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{φ}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{θ}}}$ (8)

Value

 >
 $\frac{\frac{{1}}{{2}}{}{\mathrm{π}}{-}\frac{{1}}{{4}}{}\sqrt{{3}}{}{\mathrm{π}}}{\frac{{2}}{{3}}{}{\mathrm{π}}{-}\frac{{1}}{{3}}{}\sqrt{{3}}{}{\mathrm{π}}}$ (9)



 Commands Used