Find the average value of the function over the region.
-$\mathrm { f } ( \mathrm { Q } , \varphi , \theta ) = \varrho$ over the solid ball $\varrho \leq 18$
A) $\frac { 27 } { 2 }$
B) $\frac { 2 } { 27 }$
C) $18 \pi$
D) 18

Question

Quizplus · Accepted Answer

The average value of a function $f(\rho, \phi, \theta) = \rho$ over a solid ball of radius $R$ (in this case, $R = 18$) in spherical coordinates is given by the formula $\frac{3}{4\pi R^3} \int_{0}^{R} \rho \cdot 4\pi \rho^2 d\rho$. Integrating $\rho^3$ from 0 to 18 and multiplying by $\frac{3}{4\pi (18)^3}$ gives the average value of $\frac{27}{2}$.

Find the Average Value of the Function Over the Region f(Q,φ,θ)=ϱ\mathrm { f } ( \mathrm { Q } , \varphi , \theta ) = \varrhof(Q,φ,θ)=ϱ

Find the Average Value of the Function Over the Region $\mathrm { f } ( \mathrm { Q } , \varphi , \theta ) = \varrho$