Find the average value of the function over the region.
-$f ( \varrho , \varphi , \theta ) = \varrho \cos \varphi$ over the solid lower ball $\varrho \leq 8 , \pi / 2 \leq \varphi \leq \pi$
A) $\frac { 1 } { 3 }$
B) $\frac { 3 } { 2 }$
C) $- 3$
D) $- 1024 \pi$

Question

Quizplus · Accepted Answer

The average value of the function is calculated by integrating over the given region and dividing by the volume of the region. The integral of $\varrho \cos \varphi$ over the specified region results in $-3072\pi$, and the volume of the region is $1024\pi$. Dividing these gives the average value of $-3$.

Find the Average Value of the Function Over the Region f(ϱ,φ,θ)=ϱcos⁡φf ( \varrho , \varphi , \theta ) = \varrho \cos \varphif(ϱ,φ,θ)=ϱcosφ

Find the Average Value of the Function Over the Region $f ( \varrho , \varphi , \theta ) = \varrho \cos \varphi$