Find the average value of the function over the region.
-$f ( r , \theta , z ) = r ^ { 2 }$ over the region bounded by the cylinder $r = 3$ between the planes $z = - 7$ and $z = 7$
A) 2
B) $\frac { 2 } { 9 }$
C) $\frac { 9 } { 2 }$
D) $\frac { 567 } { 2 } \pi$

Question

Quizplus · Accepted Answer

The average value of a function over a region is given by the formula:

$$\frac{1}{V}\iiint\limits_{V}f(r,	heta,z)dV$$

where $V$ is the volume of the region.

In this case, the region is the portion of the cylinder $r=3$ between the planes $z=-7$ and $z=7$, so we have:

$$V=\int_{-7}^{7}\int_{0}^{2\pi}\int_{0}^{3}rdrd	heta dz=2\pi\int_{0}^{3}rdr\int_{-7}^{7}dz=\frac{243}{2}\cdot14\pi=\frac{1701}{2}\pi$$

and

$$\iiint\limits_{V}f(r,	heta,z)dV=\int_{-7}^{7}\int_{0}^{2\pi}\int_{0}^{3}r^3drd	heta dz=\frac{81}{4}\cdot14\pi=\frac{567}{2}\pi$$

Therefore, the average value of the function is:

$$\frac{1}{V}\iiint\limits_{V}f(r,	heta,z)dV=\frac{\frac{567}{2}\pi}{\frac{1701}{2}\pi}=\frac{9}{2}$$

So the answer is C.

Find the Average Value of the Function Over the Region f(r,θ,z)=r2f ( r , \theta , z ) = r ^ { 2 }f(r,θ,z)=r2

Find the Average Value of the Function Over the Region $f ( r , \theta , z ) = r ^ { 2 }$