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Calculus - Multivariate≻Integration≻Average Value≻Polar

Table 6.4.10(a)   In polar coordinates, computation of average value by task template

Initialize

$F\=r\+\cos \left(\mathrm{\θ}\right)$$\stackrel{\text{assign}}{\to }$

Calculate the average value of $F$ 

$\frac{{\int }_{-\mathrm{\π}\/3}^{\mathrm{\π}\/3}{\int }_{2- \cos \left(\mathrm{\θ}\right)}^{3 \cos \left( \mathrm{\θ}\right)}F\cdot r \mathit{\ⅆ}r \mathit{\ⅆ}\mathrm{\θ}}{{\int }_{-\mathrm{\π}\/3}^{\mathrm{\π}\/3}{\int }_{2- \cos \left(\mathrm{\θ}\right)}^{3 \cos \left( \mathrm{\θ}\right)}r \mathit{\ⅆ}r \mathit{\ⅆ}\mathrm{\θ}}$ = $\frac{1}{9}\left(12\sqrt{3}-\frac{16}{9}\mathrm{\π}\right)\sqrt{3}$$\stackrel{\text{expand}}{\=}$$4-\frac{16}{81}\sqrt{3}\mathrm{\π}$$\stackrel{\text{at 10 digits}}{\to }$$2.925155932$

Table 6.4.10(b)   From first principles and in polar coordinates, calculation of average value

Initialize

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

Compute the average value of $F$ 

$\mathrm{Student}:-\mathrm{MultivariateCalculus}:-\mathrm{FunctionAverage}\left(F\,r\=2- \cos \left(\mathrm{\θ}\right)..3 \cos \left(\mathrm{\θ}\right)\,\mathrm{\θ}\=-\mathrm{\π}\/3..\mathrm{\π}\/3\,\mathrm{coordinates}\=\mathrm{polar}\left[r\,\mathrm{\θ}\right]\,\mathrm{output}\=\mathrm{integral}\right)$

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

$q≔\mathrm{Student}:-\mathrm{MultivariateCalculus}:-\mathrm{FunctionAverage}\left(F\,r\=2- \cos \left(\mathrm{\theta }\right)..3 \cos \left(\mathrm{\theta }\right)\,\mathrm{\θ}\=-\mathrm{\π}\/3..\mathrm{\π}\/3\,\mathrm{coordinates}\=\mathrm{polar}\left[r\,\mathrm{\theta }\right]\right)$

$\frac{1}{9}\left(12\sqrt{3}-\frac{16}{9}\mathrm{\π}\right)\sqrt{3}$

$\mathrm{expand}\left(q\right)$ = $4-\frac{16}{81}\sqrt{3}\mathrm{\π}$

$\mathrm{evalf}\left(q\right)$ = $2.925155934$