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Calculus - Multivariate≻Integration≻Average Value≻Polar

Table 6.4.6(a)   In polar coordinates, computation of average value by task template

Initialize

$F\=1\+2 r \cos \left(\mathrm{\theta }\right)\+3 r \sin \left(\mathrm{\theta }\right)$$\stackrel{\text{assign}}{\to }$

Calculate the average value of $F$ 

$\frac{{\int }_{-\mathrm{\pi }\/4}^{\mathrm{\pi }\/4}{\int }_0^{\cos \left(2 \mathrm{\theta }\right)}F\cdot r \mathit{\ⅆ}r \mathit{\ⅆ}\mathrm{\θ}}{{\int }_{-\mathrm{\π}\/4}^{\mathrm{\π}\/4}{\int }_0^{\cos \left(2 \mathrm{\θ}\right)}r \mathit{\ⅆ}r \mathit{\ⅆ}\mathrm{\θ}}$ = $\frac{8\left(\frac{32}{105}\sqrt{2}\+\frac{1}{8}\mathrm{\π}\right)}{\mathrm{\π}}$$\stackrel{\text{expand}}{\=}$$\frac{256}{105}\frac{\sqrt{2}}{\mathrm{\π}}\+1$$\stackrel{\text{at 10 digits}}{\to }$$2.097528461$$$

Table 6.4.6(b)   From first principles and in polar coordinates, calculation of average value

Initialize

$F≔1\+2 r \cos \left(\mathrm{\θ}\right)\+3 r \sin \left(\mathrm{\θ}\right)\:$

Compute the average value of $F$ 

$\mathrm{Student}:-\mathrm{MultivariateCalculus}:-\mathrm{FunctionAverage}\left(F\,r\=0..\cos \left(2 \mathrm{\θ}\right)\,\mathrm{\θ}\=-\mathrm{\π}\/4..\mathrm{\π}\/4\,\mathrm{coordinates}\=\mathrm{polar}\left[r\,\mathrm{\θ}\right]\,\mathrm{output}\=\mathrm{integral}\right)$

$\frac{{\∫}_{-\frac{1}{4}\mathrm{\π}}^{\frac{1}{4}\mathrm{\π}}{\∫}_0^{\cos \left(2\mathrm{\θ}\right)}\left(1\+2r\cos \left(\mathrm{\θ}\right)\+3r\sin \left(\mathrm{\θ}\right)\right)r\ⅆr\ⅆ\mathrm{\θ}}{{\∫}_{-\frac{1}{4}\mathrm{\π}}^{\frac{1}{4}\mathrm{\π}}{\∫}_0^{\cos \left(2\mathrm{\θ}\right)}r\ⅆr\ⅆ\mathrm{\θ}}$

$q≔\mathrm{Student}:-\mathrm{MultivariateCalculus}:-\mathrm{FunctionAverage}\left(F\,r\=0..\cos \left(2 \mathrm{\theta }\right)\,\mathrm{\theta }\=-\mathrm{\pi }\/4..\mathrm{\pi }\/4\,\mathrm{coordinates}\=\mathrm{polar}\left[r\,\mathrm{\theta }\right]\right)$

$\frac{8\left(\frac{32}{105}\sqrt{2}\+\frac{1}{8}\mathrm{\π}\right)}{\mathrm{\π}}$

$\mathrm{expand}\left(q\right)$ = $\frac{256}{105}\frac{\sqrt{2}}{\mathrm{\π}}\+1$$$

$\mathrm{evalf}\left(q\right)$ = $2.097528462$$$