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Average Value of a Function in Polar Coordinates



 Description In the polar coordinate system, where the point $\left(x,y\right)$ has coordinates , determine the average value of a function.

Average Value of a Function in Polar Coordinates

Integrand

 > $\frac{\mathrm{θ}}{{1}{+}{{r}}^{{2}}}$
 $\frac{{\mathrm{θ}}}{{1}{+}{{r}}^{{2}}}$ (1)

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

${r}_{1}\left(\mathrm{θ}\right)$

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

${r}_{2}\left(\mathrm{θ}\right)$

 > ${\mathrm{\theta }}$
 ${\mathrm{θ}}$ (3)

$a$

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

$b$

 > $\frac{{\mathrm{\pi }}}{{3}}$
 $\frac{{1}}{{3}}{}{\mathrm{π}}$ (5)

Inert Integral:

(Note automatic insertion of Jacobian.)

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

Value

 > $\mathrm{Student}\left[\mathrm{MultivariateCalculus}\right]\left[\mathrm{FunctionAverage}\right]\left(,{r}=..,{\mathrm{θ}}=..,\mathrm{coordinates}=\mathrm{polar}\left[{r}{,}{\mathrm{θ}}\right]\right)$
 $\frac{{162}{}\left({-}\frac{{1}}{{2}}{}{\mathrm{ln}}{}\left({3}\right){+}\frac{{1}}{{4}}{}{\mathrm{ln}}{}\left({9}{+}{{\mathrm{π}}}^{{2}}\right){-}\frac{{1}}{{18}}{}{{\mathrm{π}}}^{{2}}{}{\mathrm{ln}}{}\left({3}\right){+}\frac{{1}}{{36}}{}{{\mathrm{π}}}^{{2}}{}{\mathrm{ln}}{}\left({9}{+}{{\mathrm{π}}}^{{2}}\right){-}\frac{{1}}{{36}}{}{{\mathrm{π}}}^{{2}}\right)}{{{\mathrm{π}}}^{{3}}}$ (7)



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