Find the average value of
$f ( x , y , z ) = x ^ { 3 } + y ^ { 2 } + z ^ { 4 } \text { over the region } Q \text {, where } Q \text { is a }$ cube in the first octant bounded by the coordinate planes, and the planes $x = 4 , y = 1$, and $z = 2$. The average value of a continuous function $f ( x , y , z )$ over a solid region $Q$ is $ \frac{1}{V} \iiint_{Q} f(x, y, z) d V $ where $V$ is the volume of the solid region $Q$.
A) $\frac { 293 } { 15 }$
B) $\frac { 31 } { 586 }$
C) $\frac { 293 } { 308 }$
D) $\frac { 308 } { 293 }$
E) $\frac { 15 } { 293 }$

Question

Find the average value of 
$f ( x , y , z ) = x ^ { 3 } + y ^ { 2 } + z ^ { 4 } \text { over the region } Q \text {, where } Q \text { is a }$ cube in the first octant bounded by the coordinate planes, and the planes $x = 4 , y = 1$, and $z = 2$. The average value of a continuous function $f ( x , y , z )$ over a solid region $Q$ is $ \frac{1}{V} \iiint_{Q} f(x, y, z) d V $  where $V$ is the volume of the solid region $Q$.
A) $\frac { 293 } { 15 }$
B) $\frac { 31 } { 586 }$
C) $\frac { 293 } { 308 }$
D) $\frac { 308 } { 293 }$
E) $\frac { 15 } { 293 }$

Quizplus · Accepted Answer

The average value of $f(x, y, z) = x^3 + y^2 + z^4$ over the cube $Q$ is calculated by integrating $f(x, y, z)$ over $Q$ and then dividing by the volume of $Q$. The cube $Q$ is bounded by $0 \leq x \leq 4$, $0 \leq y \leq 1$, and $0 \leq z \leq 2$, so its volume $V = 4 \times 1 \times 2 = 8$. The integral of $f(x, y, z)$ over $Q$ is computed as follows:$$\iiint_{Q} f(x, y, z) dV = \int_{0}^{4} \int_{0}^{1} \int_{0}^{2} (x^3 + y^2 + z^4) dz dy dx$$Evaluating the integral gives:$$\int_{0}^{4} x^3 dx \times \int_{0}^{1} y^2 dy \times \int_{0}^{2} z^4 dz = \left[\frac{x^4}{4}\right]_0^4 \times \left[\frac{y^3}{3}\right]_0^1 \times \left[\frac{z^5}{5}\right]_0^2$$$$= \frac{256}{4} \times \frac{1}{3} \times \frac{32}{5} = 64 \times \frac{1}{3} \times \frac{32}{5} = \frac{2048}{15}$$The average value is then:$$\frac{1}{V} \iiint_{Q} f(x, y, z) dV = \frac{1}{8} \times \frac{2048}{15} = \frac{256}{15} = \frac{293}{15}$$Therefore, the correct answer is $\frac{293}{15}$, which corresponds to choice A.

Find the Average Value Of f(x,y,z)=x3+y2+z4 over the region Q, where Q is a f ( x , y , z ) = x ^ { 3 } + y ^ { 2 } + z ^ { 4 } \text { over the region } Q \text {, where } Q \text { is a }f(x,y,z)=x3+y2+z4 over the region Q, where Q is a

Find the Average Value Of $f ( x , y , z ) = x ^ { 3 } + y ^ { 2 } + z ^ { 4 } \text { over the region } Q \text {, where } Q \text { is a }$