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Average value of a function over a domain in space.

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The ave...
...rset{\mathcal{D}}{\int \int \int f(x,y,z,) \;dV}
\end{equation*}\end{minipage} }

Example 7.14   Compute the average of the function f(x,y,z)=x+y+z over the box defined by $0 \leq x \leq 1$, $0 \leq y \leq 3$ and $0 \leq z \leq 2$.

Figure 10: A box in 3-dimensional space.
\mbox{\epsfig{file=box232.eps,height=5cm} }

The volume of the box is (trivially) equal to 12. Let us compute the triple integral:

\begin{align*}I &= \int_0^2 \int_0^3 \int_0^2 (x+y+z) \; dx \; dy \; dz \\
\quad &=\int_0^2 (9+6z) \; dz \\
\quad &= [ 9z + 3z^2 ]_0^2 = 30.

\begin{displaymath}\mu_{f,\mathcal{D}} = \frac {30}{12} = \frac 52.

If the given function describes, say, the density of a material, the average is the average density over the box.

Noah Dana-Picard