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Calculus 3 (Math 221) Notes and Study Guide

Brenton LeMesurier

Section 5.4 Triple Integrals

Updated 2025-04-16, adding Subsection 5.4.5 on Type 1, 2 and 3 Regions.

References.

Introduction.

The ideas used to define double integrals and then evaluate then in terms of iterated integrals can be extended to triple integrals for functions of three variables: as usual, getting from one dimension to two has taken care of most the hard work and new ideas.

Topics.

Subsection 5.4.1 Triple Integrals over a Box

In place of rectangles in the plane, we start with integrals over boxes in space:
\begin{equation*} B = \{(x,y,z) | a \leq x \leq b, c \leq y \leq d, r \leq z \leq s \} = [a,b] \times [c,d] \times [r,s] \end{equation*}
Then we divide this into many small boxes, by first dividing the ranges of \(x\text{,}\) \(y\) and \(z\) values into sub-intervals.
Using \(l\) \(x\)-subintervals of length \(\Delta x = (b-a)/l\text{,}\) \(m\) \(y\)-subintervals of length \(\Delta y = (d-c)/m\text{,}\) \(n\) \(z\)-subintervals of length \(\Delta z = (s-r)/n\text{,}\) the boxes are
\begin{equation*} B_{ijk} = [x_{i-1},x_i] \times [y_{j-1},y_j] \times [z_{k-1},z_k] \end{equation*}
with
\begin{equation*} x_i = a + i \Delta x, y_j = c + j \Delta y, z_k = r + k \Delta z. \end{equation*}
Choosing a point \((x_{ijk}^*,y_{ijk}^*,z_{ijk}^*)\) in each box, the integral is approximated by the triple Riemann sum for function \(f\) on box \(B\text{:}\)
\begin{equation*} \sum_{i=1}^l \sum_{j=1}^m \sum_{k=1}^n f((x_{ijk}^*,y_{ijk}^*,z_{ijk}^*) \Delta V, \quad \Delta V = \Delta x \Delta y \Delta z. \end{equation*}
One natural choice of the evaluation points is again the mid-points
\begin{equation*} (x_{ijk}^*,y_{ijk}^*,z_{ijk}^*) = (\bar{x}_i,\bar{y}_j,\bar{z}_k) = \left(\frac{x_{i-1}+x_i}2, \frac{y_{j-1}+y_j}2, \frac{z_{k-1}+z_k}2\right). \end{equation*}

Definition 5.4.1. Triple Integral over a Box.

The triple integral of \(f\) over box \(B\) is
\begin{equation*} \iiint\limits_B f(x,y,z)\, dV = \lim_{l,m,n \to \infty} \sum_{i=1}^l \sum_{j=1}^m \sum_{k=1}^n f((x_{ijk}^*,y_{ijk}^*,z_{ijk}^*) \Delta V\text{,} \end{equation*}
if this limit exists.
As you can probably guess, triple integrals can be expressed in terms of three nested integrals, over the three variables:
For most practical purposes, this iterated integral could be used as the definition of the triple integral over a rectangular box.

Subsection 5.4.2 Triple Integrals over Bounded Regions

For a region \(E\) in \(\mathbb{R}^3\) that is bounded, and with its boundary a finite collection of smooth surfaces, we can define the triple integral with the same method as in Subsection 5.2.1: define the function outside region \(E\) to have value zero, and integrate this extended function over a box containing \(E\text{.}\)
In practice, the most manageable regions are ones akin to the Type I and Type II regions in the plane. Just as Type I regions were described as "Type \(dy\)-\(dx\)" in Subsection 5.2.2, indicating the order in which the iterated integrals can easily be done, this jargon can be extended to three dimensional regions as "Type \(dz\)-\(dy\)-\(dx\)" (as with the order in Equation (5.4.1)) and so on.
That is, a type \(dz\)-\(dy\)-\(dx\) region E is one of the form
\begin{equation} E = \{(x,y,z) | a \leq x \leq b, g_1(x) \leq y \leq g_2(x), u_1(x,y) \leq z \leq u_2(x,y) \}\tag{5.4.2} \end{equation}
(As described in the text, this is a type I region in space, sitting over a type I region \(D\) in the plane.)

Subsection 5.4.3 Iterated Integral Form for Type \(dz\)-\(dy\)-\(dx\) Regions in Space

The triple integral over such a region is given by
\begin{equation*} \iiint\limits_E f(x,y,z) \,dV = \int_{x=a}^b \int_{y=g_1(x)}^{g_2(x)} \int_{z=u_1(x,y)}^{u_2(x,y)} f(x,y,z) \,dz \,dy \,dx \end{equation*}

Example 5.4.3. Integration over a Ball.

The ball \(E= \{(x,y,z) | x^2 + y^2 + z^2 \leq R^2 \}\text{,}\) can be described by the inequalities \(\ds -R \leq x \leq R\text{,}\) \(-\sqrt{1-x^2} \leq y \leq \sqrt{1-x^2}\text{,}\) \(-\sqrt{1-x^2-y^2} \leq z \leq \sqrt{1-x^2-y^2}\text{.}\)
Thus, the integral over such a ball is
\begin{equation*} \iiint\limits_{x^2+y^2+z^2 \leq R^2} \kern-2ex f(x,y,z) \,dV = \int\limits_{x=-R}^R \int\limits_{y=-\sqrt{R^2-x^2}}^{\sqrt{R^2-x^2}} \int\limits_{z=-\sqrt{R^2-x^2-y^2}}^{\sqrt{R^2-x^2-y^2}} f(x,y,z) \,dz \,dy \,dx \end{equation*}
(or any of five similar reorderings).

Subsection 5.4.4 Changing the Order of Integration

With iterated triple integrals there are six possible order for the integrals, and just as seen with Type I and Type II regions for double integrals, not all are possible for all domains. Also, some choices of order might make the differnce between an easier integral, a harder one, or one that cannot not be evaluated using known antiderivatives.
Thus, it can be useful to seek to change the order of integrals, and again one key is solving equations for the boundary of the domain, which then give the inequalities describing the limits of each integral.
An important point is that, as in Equation (5.4.2) and in Equations (5.2.2) and (5.2.3) fot Type I and Type II double integrals, the limits of integration of each integral might depend on the variable of an integral "outside" it (integral sign to the left) but cannot refer to the variable in another integral "inside it" (integral sign to the right).

Subsection 5.4.5 Type 1, 2 and 3 Regions

Although the above six region types are convenient for setting up iterated integral, many results later can be got with just three types, depending on which variable is integrated first, by “boot-strapping” on what we already know about double integrals.
Define a “Type 1” region is one of the form
\begin{equation} E = \{(x, y, z): (x,y) \in D, B(x,y) \le z \leq T(x,y)\text{,}\tag{5.4.3} \end{equation}
analogous to a Type I region in \(\reals^2\text{.}\) For these,
\begin{equation} \iiint\limits_E f(x, y, z)\ dV = \iint\limits_D \int_{B(x,y)}^{T(x,y)} f(x, y, z) dz\ dV \text{.}\tag{5.4.4} \end{equation}
(The other two types are defined similarly, but we only really need to talk in detail about this version.)

Subsection 5.4.6 Volumes and Averages

Just as area can be computed in terms of a double integral, a bounded region \(E\) in \(\mathbb{R}^3\) has volume
\begin{equation*} V(E) = \iiint\limits_E 1\ dV, \text{ often abbreviated } \iiint\limits_E\ dV \end{equation*}
and it should be no surprise that the average of a function \(f(x,y,z)\) over region \(E\) is defined as
\begin{equation*} \bar{f} = \frac{\iiint_E f(x,y,z)\, dV}{V(E)}. \end{equation*}

Study Guide.

Study Section 5.4 of Calculus Volume 3
 6 
openstax.org/books/calculus-volume-3/pages/5-4-triple-integrals
; in particular
  • The definition of a triple integral.
  • Fubini’s Theorem for triple Integrals (Theorem 5.9), expressing triple integrals as iterated integrals.
  • Theorem 5.10, expressing a triple integral over a general region in terms of iterated integrals.
  • Examples 36–39 and above all, Examples 40 and 41. (Plus as usual the Checkpoints following each).
  • One or several exercises from each of the following ranges: 181–184, 185–188, 191–194, 195–198, and 211 and/or 212.
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