Tools≻Tasks≻Browse:

Calculus - Multivariate≻Integration≻Average Value≻Polar

Table 6.4.8(a)   In polar coordinates, computation of average value by task template

Initialize

$F\=r^2$$\stackrel{\text{assign}}{\to }$

Calculate the average value of $F$ 

$\frac{{\int }_{5 \mathrm{\π}\/6}^{\mathrm{\π}\/6}{\int }_{1\+\sin \left(\mathrm{\θ}\right)}^{3 \sin \left( \mathrm{\θ}\right)}F\cdot r \mathit{\ⅆ}r \mathit{\ⅆ}\mathrm{\θ}}{{\int }_{5 \mathrm{\π}\/6}^{\mathrm{\π}\/6}{\int }_{1\+\sin \left(\mathrm{\θ}\right)}^{3 \sin \left( \mathrm{\θ}\right)}r \mathit{\ⅆ}r \mathit{\ⅆ}\mathrm{\θ}}$ = $-\frac{-\frac{9}{4}\sqrt{3}-\frac{13}{3}\mathrm{\π}}{\mathrm{\π}}$$\stackrel{\text{expand}}{\=}$$\frac{9}{4}\frac{\sqrt{3}}{\mathrm{\π}}\+\frac{13}{3}$$\stackrel{\text{at 10 digits}}{\to }$$5.573823348$

Table 6.4.8(b)   From first principles and in polar coordinates, calculation of average value

Initialize

$F≔r^2\:$

Compute the average value of $F$ 

$\mathrm{Student}:-\mathrm{MultivariateCalculus}:-\mathrm{FunctionAverage}\left(F\,r\=1\+\sin \left(\mathrm{\θ}\right)..3 \sin \left(\mathrm{\θ}\right)\,\mathrm{\θ}\=\mathrm{\π}\/6..5 \mathrm{\π}\/6\,\mathrm{coordinates}\=\mathrm{polar}\left[r\,\mathrm{\θ}\right]\,\mathrm{output}\=\mathrm{integral}\right)$

$\frac{{\∫}_{\frac{1}{6}\mathrm{\π}}^{\frac{5}{6}\mathrm{\π}}{\∫}_{1\+\sin \left(\mathrm{\θ}\right)}^{3\sin \left(\mathrm{\θ}\right)}{r}^3\ⅆr\ⅆ\mathrm{\θ}}{{\∫}_{\frac{1}{6}\mathrm{\π}}^{\frac{5}{6}\mathrm{\π}}{\∫}_{1\+\sin \left(\mathrm{\θ}\right)}^{3\sin \left(\mathrm{\θ}\right)}r\ⅆr\ⅆ\mathrm{\θ}}$

$q≔\mathrm{Student}:-\mathrm{MultivariateCalculus}:-\mathrm{FunctionAverage}\left(F\,r\=1\+\sin \left(\mathrm{\theta }\right)..3 \sin \left(\mathrm{\theta }\right)\,\mathrm{\θ}\=\mathrm{\π}\/6..5 \mathrm{\π}\/6\,\mathrm{coordinates}\=\mathrm{polar}\left[r\,\mathrm{\theta }\right]\right)$

$\frac{\frac{9}{4}\sqrt{3}\+\frac{13}{3}\mathrm{\π}}{\mathrm{\π}}$

$\mathrm{expand}\left(q\right)$ = $\frac{9}{4}\frac{\sqrt{3}}{\mathrm{\π}}\+\frac{13}{3}$$$

$\mathrm{evalf}\left(q\right)$ = $5.573823350$$$