5.4 Triple integrals  (Page 5/8)

Average value of a function of three variables

Recall that we found the average value of a function of two variables by evaluating the double integral over a region on the plane and then dividing by the area of the region. Similarly, we can find the average value of a function in three variables by evaluating the triple integral over a solid region and then dividing by the volume of the solid.

Key concepts

• To compute a triple integral we use Fubini’s theorem, which states that if $f\left(x,y,z\right)$ is continuous on a rectangular box $B=\left[a,b\right]\times \left[c,d\right]\times \left[e,f\right],$ then
  
  ${\displaystyle \underset{B}{\iiint }f\left(x,y,z\right)dV={\displaystyle \underset{e}{\overset{f}{\int }}{\displaystyle \underset{c}{\overset{d}{\int }}{\displaystyle \underset{a}{\overset{b}{\int }}f\left(x,y,z\right)dxdydz}}}}$
  
  and is also equal to any of the other five possible orderings for the iterated triple integral.
• To compute the volume of a general solid bounded region $E$ we use the triple integral
  
  $V\left(E\right)={\displaystyle \underset{E}{\iiint }1dV}.$
• Interchanging the order of the iterated integrals does not change the answer. As a matter of fact, interchanging the order of integration can help simplify the computation.
• To compute the average value of a function over a general three-dimensional region, we use
  
  $f_{\text{ave}}=\frac{1}{V\left(E\right)}{\displaystyle \underset{E}{\iiint }f\left(x,y,z\right)}dV.$

Key equations

• Triple integral
  $\underset{l,m,n\to \infty }{\text{lim}}{\displaystyle \sum _{i=1}^l{\displaystyle \sum _{j=1}^m{\displaystyle \sum _{k=1}^{n}f(x_{ijk}^{*},y_{ijk}^{*},z_{ijk}^{*})\text{Δ}x\text{Δ}y\text{Δ}z}}}={\displaystyle \underset{B}{\iiint }f(x,y,z)dV}$

In the following exercises, evaluate the triple integrals over the rectangular solid box $B.$

In the following exercises, change the order of integration by integrating first with respect to $z,$ then $x,$ then $y.$