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Active Calculus - Multivariable

Nicholas Long, Mitchel T. Keller, Steve Schlicker

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Section 12.6 Triple Integrals

Notes to the Instructor and Dependencies.

This section extends the tools of Section 12.3 to functions of three variables to define triple integrals. While these ideas are geometrically challenging to students (finding bounds of integration and visualization of volumes in three dimensions), many students understand the algebraic side of this section quickly and without much prompting.
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Subsection 12.6.1 Introduction

In the first five sections of this chapter, we defined the double integral of a continuous function \(f = f(x,y)\) over a rectangle \(R = [a,b] \times [c,d]\) as a limit of a double Riemann sum, which paralleled the single-variable integral of a function \(g = g(x)\) on an interval \([a,b]\text{.}\) We have also repeatedly emphasized the interpretations and applications of the double integral as stated in Key Idea 12.1.8. It is natural to wonder if it is possible to extend these ideas of double Riemann sums and double integrals for functions of two variables to triple Riemann sums and then triple integrals for functions of three variables. We begin investigating this generalization in Preview Activity 12.6.1 by looking at setting up the classical calculus approach to find the mass of a rectangular solid region of varying density in three dimensions.
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Preview Activity 12.6.1.
Consider a solid piece of granite in the shape of a box described by
\begin{equation*} B = \{(x,y,z) : 0 \leq x \leq 4, 0 \leq y \leq 6, 0 \leq z \leq 8\}\text{.} \end{equation*}
Suppose that the density of the granite varies from point to point. Let \(\delta(x, y, z)\) represent the mass density of the piece of granite at point \((x,y,z)\) in kilograms per cubic meter. (Note that this means that \(x\text{,}\) \(y\text{,}\) and \(z\) are measured in meters). Our goal is to find the mass of this solid.
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For a solid of constant density, we can find the mass by multiplying the density and volume. Here, the density varies from point to point. Therefore, we will use the approach we did with two-variable lamina problems. We will apply the first step of the classic calculus approach by slicing the solid into smaller pieces on which the density changes very little and thus is close to constant. We will use these pieces to approximate the total mass.
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(a)
For a first approximation using smaller pieces, we partition the interval \([0,4]\) into two subintervals of equal length, the interval \([0,6]\) into three subintervals of equal length, and the interval \([0,8]\) into two subintervals of equal length. This partitions the box \(B\) into sub-boxes as shown in Figure 12.6.1.
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A plot of a box-shaped region in three dimensions. The box has one corner at the origin and the diagonally opposite corner is at \((4,6,8)\text{.}\) The box is partitioned into smaller boxes by one horizontal plane, one vertical plane parallel to the \(yz\)-plane, and two vertical planes parallel to the \(xz\)-plane.
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Figure 12.6.1. A partitioned three-dimensional domain
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Let \(0=x_0 \lt x_1 \lt x_2=4\) be the endpoints of the subintervals of \([0,4]\) after partitioning. Label these endpoints on Figure 12.6.1. Repeat this process with \(0=y_0 \lt y_1 \lt y_2 \lt y_3=6\) and \(0=z_0 \lt z_1 \lt z_2=8\text{.}\) Find the length \(\Delta x\) of each subinterval \([x_{i-1},x_i]\) as well as the lengths \(\Delta y\) and \(\Delta z\) of the subintervals along the \(y\)- and \(z\)-axes, respectively.
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(b)
The partitions of the intervals \([0,4]\text{,}\) \([0,6]\) and \([0,8]\) partition the box \(B\) into sub-boxes. Find the number of sub-boxes as well as the volume \(\Delta V\) of each sub-box.
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(c)
Let \(B_{ijk}\) denote the sub-box \([x_{i-1},x_i] \times [y_{j-1},y_j] \times [z_{k-1}, z_k]\text{.}\) Suppose that for each possible combination of \(i,j,k\text{,}\) we choose a point \((x_{ijk}^*, y_{ijk}^*, z_{ijk}^*)\) in \(B_{ijk}\) . Write a couple of sentences to describe what physical quantity \(\delta(x_{ijk}^*, y_{ijk}^*, z_{ijk}^*) \Delta V\) approximates. Be sure to explain the meaning of \(\delta(x_{ijk}^*, y_{ijk}^*, z_{ijk}^*)\) in your response.
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(d)
We partitioned the space of inputs for \(\delta(x, y, z)\) into a relatively small number of pieces. Write a couple of sentences about what you would change in the process above to improve the approximation of the mass of the box. Be sure to explain why your changes will give a better approximation.
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In the Preview Activity, we completed most of the classic calculus approach to set up integration of a density function in three dimensions. In the next subsection, we will generalize these ideas and work on activities and examples that highlight the definition of a triple integral and how to convert triple integrals to iterated integrals for efficient calculation.
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Subsection 12.6.2 Triple Riemann Sums and Triple Integrals

Through the application of a mass density distribution over a three-dimensional solid, Preview Activity 12.6.1 suggests a natural generalization from double Riemann sums of functions of two variables to triple Riemann sums of functions of three variables. In the same way, we can generalize from double integrals to triple integrals. By simply adding a \(z\)-coordinate to our earlier work, we can define both a triple Riemann sum and the corresponding triple integral.
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Definition 12.6.2.

Let \(f = f(x,y,z)\) be a continuous function on a box \(B = [a,b] \times [c,d] \times [r,s]\text{.}\) The triple Riemann sum of \(f\) over \(B\) is created as follows.
  • Partition the interval \([a,b]\) into \(m\) subintervals of equal length \(\Delta x = \frac{b-a}{m}\text{.}\) Let \(x_0=a,x_1,\ldots,x_m=b\) be the endpoints of these subintervals.
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  • Partition the interval \([c,d]\) into \(n\) subintervals of equal length \(\Delta y = \frac{d-c}{n}\text{.}\) Let \(y_0=c,y_1,\ldots,y_m=d\) be the endpoints of these subintervals.
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  • Partition the interval \([r,s]\) into \(\ell\) subintervals of equal length \(\Delta z = \frac{s-r}{\ell}\text{.}\) Let \(z_0=r,z_1,\ldots,z_\ell=s\) be the endpoints of these subintervals.
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  • For \(i\) between \(1\) and \(m\text{,}\) \(j\) between \(1\) and \(n\text{,}\) and \(k\) between 1 and \(\ell\text{,}\) let \(B_{ijk}\) be the sub-box of \(B\) with diagonally-opposite vertices \((x_{i-1},y_{j-1},z_{k-1})\) and \((x_i, y_j, z_k)\) . The volume of each \(B_{ijk}\) is
    \begin{equation*} \Delta V = \Delta x \cdot \Delta y \cdot \Delta z\text{.} \end{equation*}
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  • For each \(i\text{,}\) \(j\text{,}\) and \(k\text{,}\) let \((x_{ijk}^*, y_{ijk}^*, z_{ijk}^*)\) be a point in box \(B_{ijk}\text{.}\) The resulting triple Riemann sum for \(f\) on \(B\) is
    \begin{equation*} \sum_{i=1}^m \sum_{j=1}^n \sum_{k=1}^{\ell} f(x_{ijk}^*, y_{ijk}^*, z_{ijk}^*) \cdot \Delta V\text{.} \end{equation*}
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If \(f(x,y,z)\) represents the mass density of a material in the box \(B\text{,}\) then as we saw in Preview Activity 12.6.1, the triple Riemann sum approximates the total mass of material in the box \(B\text{.}\) In order to find the exact mass of the box, we need to let the number of sub-boxes increase without bound (in other words, let \(m\text{,}\) \(n\text{,}\) and \(\ell\) go to infinity). When we do this, the finite sum of the mass approximations becomes the exact mass of the solid \(B\text{.}\) More generally, we have the following definition of the triple integral.
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Definition 12.6.3.

Using the notation introduce for a triple Riemann sum, the triple integral of \(f\) over \(B\) is
\begin{equation*} \iiint_B f(x,y,z) \, dV = \lim_{m,n,\ell \to \infty} \sum_{i=1}^m \sum_{j=1}^n \sum_{k=1}^{\ell} f(x_{ijk}^*, y_{ijk}^*, z_{ijk}^*) \cdot \Delta V\text{.} \end{equation*}
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As described earlier, if \(f(x, y, z)\) represents the density of the solid \(B\) at each point \((x, y, z)\text{,}\) then
\begin{equation*} M = \iiint_B f(x,y,z) \, dV \end{equation*}
is the mass of \(B\text{.}\) Even more importantly, for any continuous function \(f\) over the solid \(B\text{,}\) we can use a triple integral to determine the average value of \(f\) over \(B\text{,}\) \(f_{\operatorname{AVG}(B)}\text{.}\) We note this generalization of our work with functions of two variables along with several others in the following important boxed information. Note that each of these quantities may actually be considered over a general domain \(S\) in \(\R^3\text{,}\) not simply a box, \(B\text{.}\)
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In rectangular coordinates, the volume element \(dV\) is \(dz \, dy \, dx\text{.}\) Thus, a triple integral of a function \(f\) over a box \(B = [a,b] \times [c,d] \times [r,s]\) in rectangular coordinates can be evaluated as an iterated integral of the form
\begin{equation*} \iiint_B f(x,y,z) \, dV = \int_a^b \int_c^d \int_r^s f(x,y,z) \, dz \, dy \, dx. \end{equation*}
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Activity 12.6.2.
Set up and evaluate an iterated integral that will evaluate the triple integral of \(f(x,y,z) = x-y+2z\) over the box \(B = [-2,3] \times [1,4] \times [0,2]\text{.}\)
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If we want to evaluate a triple integral as an iterated integral over a solid \(S\) that is not a box, then we need to describe the solid in terms of variable limits that correspond to appropriate inequalities.
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Example 12.6.5.

In this example, we will find the mass of the tetrahedron \(S\) in the first octant bounded by the coordinate planes and the plane \(x + 2 y + 3 z = 6\) if the density at point \((x,y,z)\) is given by \(\delta(x, y, z) = x + y + z\text{.}\) A picture of the solid tetrahedron is shown in Figure 12.6.6.
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The tetrahedron in three-dimensional space. Three sides of the tetrahedron are triangular regions in the coordinate planes, and the fourth side is the triangle with vertices \((6,0,0)\text{,}\) \((0,3,0)\text{,}\) and \((0,0,2)\text{.}\)
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Figure 12.6.6. The tetrahedron \(S\) in the first octant bounded by the coordinate planes and the plane \(x + 2 y + 3 z = 6\)
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We find the mass \(M\) of the tetrahedron using the triple integral
\begin{equation*} M = \iiint_S \delta(x,y,z) \, dV\text{.} \end{equation*}
To do this, we will need to generalize our ideas from Section 12.3 and describe \(S\) using three sets of inequalities, one for each variable. In this example, we choose to integrate with respect to \(z\) first for the innermost integral. Doing this requires us to first find inequalities that describe the projection of \(S\) onto the \(xy\)-plane. This region is shown in Figure 12.6.7.
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described in detail following the image
A shaded triangular region in the \(xy\)-plane. The triangle has vertices \((0,0)\text{,}\) \((6,0)\text{,}\) and \((0,3)\text{.}\) There are dashed vertical lines shown inside the region.
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Figure 12.6.7. The projection of our tetrahedron to the \(xy\)-plane, shown with vertical slices
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Looking at the region, we see that it is both vertically simple and horizontally simple, so we can describe it in either way. Here, we will describe the region as vertically simple, as suggested by the dashed lines in the image. Doing this, we see that we have slices for \(0\leq x\leq 6\text{.}\) Furthermore, the lower bound on each slice is \(y=0\text{,}\) so we just need to find the equation of the top boundary of the region. We can find the equation of the line that determines this top boundary by setting \(z = 0\) in the equation of the plane, which tells us that the resulting relation between \(x\) and \(y\) is \(x + 2 y = 6 \text{.}\) Solving for \(y\) gives \(y = 3 - \frac{1}{2}x\text{.}\) Therefore, we can describe the base of the tetrahedron as a vertically simple region using the inequalities
\begin{equation*} 0\leq x\leq 6\qquad\qquad 0 \leq y \leq 3 - \frac{1}{2}x\text{.} \end{equation*}
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To complete our inequalities describing \(S\text{,}\) we need bounds on \(z\text{.}\) These bounds can be expressed in terms of both \(x\) and \(y\) if necessary. For a fixed point \((x,y)\) in the triangular projection we have already analyzed, these bounds give the \(z\)-coordinates of the points at the bottom of \(S\) and the top of \(S\) above \((x,y)\text{.}\) The bottom of the tetrahedron is the \(xy\)-plane, so we will have \(0\leq z\text{.}\) The top of the tetrahedron is given by the equation
\begin{equation*} x + 2 y + 3 z = 6\text{.} \end{equation*}
Solving for \(z\) goves
\begin{equation*} z = \frac{1}{3}(6 - x - 2y) \end{equation*}
as the upper bound for \(z\text{.}\) Thus, we can fully describe the solid \(S\) using the inequalities
\begin{equation*} 0\leq x\leq 6\qquad\qquad 0 \leq y \leq 3 - \frac{1}{2}x \qquad\qquad 0\leq z\leq \frac{1}{3}(6 - x - 2y)\text{.} \end{equation*}
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With our description of \(S\) in terms of inequalities in hand, we can write an iterated triple integral to find the mass of the tetrahedron by integrating the density function \(\delta(x,y,z)=x+y+z\text{:}\)
\begin{equation} M = \int_{0}^{6} \int_{0}^{3-(1/2)x} \int_{0}^{(1/3)(6-x-2y)} x+y+z \, dz \, dy \, dx\text{.}\tag{12.6.1} \end{equation}
Evaluating the iterated triple integral gives us
\begin{align*} M \amp = \int_{0}^{6} \int_{0}^{3-(1/2)x} \int_{0}^{(1/3)(6-x-2y)} x+y+z \, dz \, dy \, dx\\ \amp = \int_{0}^{6} \int_{0}^{3-(1/2)x} \left[xz+yz+\frac{z}{2}\right]\restrict{0}^{(1/3)(6-x-2y)} \, dy \, dx\\ \amp = \int_{0}^{6} \int_{0}^{3-(1/2)x} \frac{4}{3}x - \frac{5}{18}x^2 - \frac{}{9}xy + \frac{2}{3}y - \frac{4}{9}y^2 + 2 \, dy \, dx\\ \amp = \int_{0}^{6} \left[\frac{4}{3}xy - \frac{5}{18}x^2y - \frac{7}{18}xy^2 + \frac{1}{3}y^2 - \frac{4}{27}y^3 + 2y \right]\restrict{0}^{3-(1/2)x} \, dx\\ \amp = \int_{0}^{6} 5 + \frac{1}{2}x - \frac{7}{12}x^2 + \frac{13}{216}x^3 \, dx\\ \amp = \left[5x + \frac{1}{4}x^2 - \frac{7}{36}x^3 + \frac{13}{864}x^4 \right] \restrict{0}^{6}\\ \amp = \frac{33}{2}. \end{align*}
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The next activity asks you to find the mass of a solid region that does not have a base lying in the \(xy\)-plane.
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Activity 12.6.3.
Let \(S\) be the solid cone bounded by \(z = \sqrt{x^2+y^2}\) and \(z=3\text{.}\) The goal of this activity is to set up an iterated integral of the form
\begin{equation} \int_{x=?}^{x=?} \int_{y=?}^{y=?} \int_{z=?}^{z=?} \delta(x,y,z) \, dz \, dy \, dx\tag{12.6.2} \end{equation}
to represent the mass of \(S\) in the setting where \(\delta(x,y,z)\) tells us the density of \(S\) at the point \((x,y,z)\text{.}\) In particular, we must find the limits on each of the three integrals.
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A picture of \(S\) is shown in Figure 12.6.8. Adjust the sliders, which will move the vertical line segment shown inside the solid. Notice that some combinations of slider values show a red point and no line segment, as the point \((x,y)\) does not correspond to any points inside the solid.
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A solid in three-dimensional space. The solid is bounded below by an upward-opening cone with its vertex at the origin. The solid is bounded above by a horizontal plane. There are sliders for \(x\) and \(y\) that control the location of a vertical line segment from the bottom of the solid to the top of the solid over the point \((x,y)\text{.}\)
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Figure 12.6.8. The region \(S\) with a vertical segment highlighted for fixed \(x\) and \(y\) values
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(a)
For the innermost integral of equation (12.6.2), we need bounds on the \(z\)-coordinate for fixed values of \(x\) and \(y\text{.}\) In Figure 12.6.8, you can use the sliders to change the values of \(x\) and \(y\text{.}\) When your choices of \(x\) and \(y\) correspond to points inside the solid, you see a vertical line segment in the plot from the bottom of the solid to the top of the solid over the point \((x,y)\) in the \(xy\)-plane.
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Try several values of \(x\) and \(y\) and look at how the length of the segment changes in the \(z\)-direction. In particular, for every \(x\) and \(y\) pair, the bottom boundary of the solid is the same. Similarly, for every pair of values the same surface is the top boundary of the solid. This allows you to state functions, in terms of \(x\) and \(y\text{,}\) that describe the top and bottom boundaries of the solid. These are the \(z\)-coordinates of the points at the top and bottom of the vertical line segments through the solid shown in the figure.
\begin{align*} z_{\text{top}}(x,y)\amp= \underline{\hspace{4cm}} \\ z_{\text{bottom}}(x,y)\amp= \underline{\hspace{4cm}} \end{align*}
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(b)
Complete the compound inequality below to provide lower and upper bounds for the \(z\)-coordinates of points in \(S\) (in terms of \(x\) and \(y\)).
\begin{equation*} \underline{\hspace{4cm}}\leq z\leq \underline{\hspace{4cm}}\text{.} \end{equation*}
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(c)
Having established upper and lower bounds for \(z\) as a function of a fixed choice of \(x\) and \(y\text{,}\) we need to describe the set of points \((x,y)\) in the \(xy\)-plane lead to vertical lines through \((x,y)\) that pass through the solid. Notice that if you choose values of \(x\) and \(y\) in Figure 12.6.8 that does not intersect the solid (e.g., \((x,y)=(2.6,-2.3)\)), then the point shown is red.
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Write a couple of sentences to explain why the set of \((x,y)\) points we need to consider is not the square \([-3,3]\times[-3,3]\text{.}\)
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(d)
On Figure 12.6.9, draw a plot of \(D\text{,}\) the region of \((x,y)\) points that correspond to points of \(S\text{.}\) We refer to \(D\) as the projection of \(S\) onto the \(xy\)-plane.
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described in detail following the image
Axes in the \(xy\)-plane. The scale on both axes goes from \(-4\) to \(4\text{.}\)
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Figure 12.6.9. A grid of \([-4,4]\times[-4,4]\)
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(e)
To complete the task of writing
\begin{equation*} \iiint_S \delta(x,y,z) dV \end{equation*}
as an iterated integral, you need to describe the region \(D\) in the \(xy\)-plane from the previous part using inequalities as with double integrals. Do this using a vertically simple description in order to have our iterated integral fit the form of equation (12.6.2).
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Give the inequalities that describe the region \(D\) from the previous part as vertically simple.
\begin{align*} \underline{\hspace{4cm}} \amp\leq x \amp\leq \underline{\hspace{4cm}} \\ \underline{\hspace{4cm}} \amp\leq y \amp\leq \underline{\hspace{4cm}} \end{align*}
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(f)
Write an iterated integral of the form (12.6.2) that represents the mass of the cone \(S\text{.}\)
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Algebraic Note: When setting up iterated integrals, the limits on a given variable can be only in terms of constants and the variable(s) whose integral(s) are inside that integral in the iterated integral. In addition, there are multiple different ways we can choose to set up such an integral. For example, two possibilities for iterated integrals that represent a triple integral \(\iiint_S f(x,y,z) \, dV\) over a solid \(S\) are
\begin{align*} \amp\int_a^b \int_{g_1(x)}^{g_2(x)} \int_{h_1(x,y)}^{h_2(x,y)} f(x,y,z) \, dz \, dy \, dx\\ \amp\int_r^s \int_{p_1(z)}^{p_2(z)} \int_{q_1(x,z)}^{q_2(x,z)} f(x,y,z) \, dy \, dx \, dz \end{align*}
where \(g_1\text{,}\) \(g_2\text{,}\) \(h_1\text{,}\) \(h_2\text{,}\) \(p_1\text{,}\) \(p_2\text{,}\) \(q_1\text{,}\) and \(q_2\) are functions of the indicated variables. There are four other options beyond the two stated here, since the variables \(x\text{,}\) \(y\text{,}\) and \(z\) can, in theory, be arranged in any order. In many circumstances, an insightful choice of variable order will make it easier to set up an iterated integral, just as was the case when we worked with double integrals.
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Setting up limits on iterated integrals often requires considerable geometric intuition. It is important to not only create carefully labeled figures, but also to think about how we wish to slice the solid. Further, note that when we say “we will integrate first with respect to \(x\text{,}\)” by “first” we are referring to the innermost integral in the iterated integral. The next activity explores several different ways we might set up the integral in the preceding example.
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Activity 12.6.4.
There are several other ways to set up the integral to give the mass of the tetrahedron in Example 12.6.5.
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(a)
How many different orders of integration could be used for iterated integrals that are equal to the integral in equation (12.6.1)?
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(b)
Set up an iterated integral, integrating first with respect to \(z\text{,}\) then \(x\text{,}\) then \(y\) that is equivalent to the integral in equation (12.6.1). Before you write down the integral, think about Figure 12.6.6, and draw an appropriate two-dimensional image of an important projection.
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(c)
Set up an iterated integral, integrating first with respect to \(y\text{,}\) then \(z\text{,}\) then \(x\) that is equivalent to the integral in equation (12.6.1). As above, think carefully about the geometry first and draw a plot of the appropriate projection.
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(d)
Set up an iterated integral, integrating first with respect to \(x\text{,}\) then \(y\text{,}\) then \(z\) that is equivalent to the integral in equation (12.6.1).
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The next activity asks you to engage in further practice with setting up triple integrals as iterated integrals, including thinking about the interpretations of Key Idea 12.6.4
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Activity 12.6.5.
A solid \(S\) is bounded below by the paraboloid \(z=x^2+y^2\) and above by the sphere \(x^2+y^2+z^2=6\text{.}\) A picture of \(S\) is shown in Figure 12.6.10.
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A three-dimensional plot of a solid region. The solid is bounded below by a surface that rises from the origin and has cicles as its horizontal cross-sections. It is bounded above by the top portion of the sphere \(x^2+y^2+z^2=6\text{.}\)
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Figure 12.6.10. The solid bounded below by the paraboloid \(z=x^2+y^2\) and above by the sphere \(x^2+y^2+z^2=6\)
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(a)
Set up an interated triple integral to find the volume of \(S\text{.}\) You do not need to evaluate the integral at this time.
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Hint.
You may find it helpful to recall the approach of Activity 12.6.3 and imagine a line segment through \(S\) as you find three compound inequalities that bound \(x\text{,}\) \(y\text{,}\) and \(z\) to get your limits of integration.
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(b)
Suppose the density at point \((x,y,z)\) is \(\delta(x,y,z)=x^2+1\text{.}\) Set up but do not evaluate iterated triple integral expressions find the center of mass of \(S\text{.}\)
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(c)
Let \(f(x,y,z) = xy+z^2\text{.}\) Set up but do not evaluate an iterated triple integral to find the average value of \(f\) on \(S\)
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(d)
Use technology appropriately to evaluate the iterated triple integrals you wrote in the other three parts of this activity. Write a couple of sentences to explain why the location of the center of mass makes sense.
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Exercises 12.6.3 Exercises

1.

Find the triple integral of the function \(f(x,y,z) = x^{2}\cos\mathopen{}\left(y+z\right)\) over the cube \(5 \leq x \leq 6\text{,}\) \(0 \leq y \leq \pi\text{,}\) \(0 \leq z \leq \pi.\)
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2.

Evaluate the triple integral
\begin{equation*} \int \!\! \int \!\! \int_{\mathbf{E}} xyz \, dV \end{equation*}
where E is the solid: \(0 \leq z \leq 8 , \ \ 0 \leq y \leq z, \ \ 0 \leq x \leq y\text{.}\)
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3.

Find the mass of the rectangular prism \(0 \leq x \leq 4, \ \ 0 \leq y \leq 3, \ \ 0 \leq z \leq 1\text{,}\) with density function \(\rho \left( x, y, z \right) = x\text{.}\)
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4.

Find the average value of the function \(f \left( x, y, z \right) = y e^{-xy}\) over the rectangular prism \(0 \leq x \leq 1\text{,}\) \(0 \leq y \leq 2\text{,}\) \(0 \leq z \leq 3\)
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5.

Find the volume of the solid bounded by the planes x = 0, y = 0, z = 0, and x + y + z = 2.
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6.

Find the mass of the solid bounded by the \(xy\)-plane, \(yz\)-plane, \(xz\)-plane, and the plane \((x/2)+(y/3)+(z/6) = 1\text{,}\) if the density of the solid is given by \(\delta (x,y,z)=x + 2 y\text{.}\)
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mass =
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7.

The moment of inertia of a solid body about an axis in 3-space relates the angular acceleration about this axis to torque (force twisting the body). The moments of inertia about the coordinate axes of a body of constant density and mass \(m\) occupying a region \(W\) of volume \(V\) are defined to be
\begin{equation*} I_x = \frac{m}{V}\int_W (y^2+z^2) \,dV\qquad I_y = \frac{m}{V}\int_W (x^2+z^2) \,dV\qquad I_z = \frac{m}{V}\int_W (x^2+y^2) \,dV \end{equation*}
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Use these definitions to find the moment of inertia about the \(z\)-axis of the rectangular solid of mass \(48\) given by \(0 \le x \le 2\text{,}\) \(0 \le y \le 2\text{,}\) \(0 \le z \le 4\text{.}\)
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\(I_x =\)
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\(I_y =\)
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\(I_z =\)
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8.

Express the integral \(\displaystyle \iiint_E f(x,y,z) dV\) as an iterated integral in six different ways, where E is the solid bounded by \(z =0, x = 0, z = y - 4 x\) and \(y = 16\text{.}\)
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1. \(\displaystyle \int_a^b \int_{g_1(x)}^{g_2(x)} \int_{h_1(x,y)}^{h_2(x,y)}f(x,y,z) dz dy dx\)
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\(a =\) \(b =\)
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\(g_1(x) =\) \(g_2(x) =\)
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\(h_1(x,y) =\) \(h_2(x,y) =\)
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2. \(\displaystyle \int_a^b \int_{g_1(y)}^{g_2(y)} \int_{h_1(x,y)}^{h_2(x,y)}f(x,y,z) dz dx dy\)
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\(a =\) \(b =\)
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\(g_1(y) =\) \(g_2(y) =\)
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\(h_1(x,y) =\) \(h_2(x,y) =\)
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3. \(\displaystyle \int_a^b \int_{g_1(z)}^{g_2(z)} \int_{h_1(y,z)}^{h_2(y,z)}f(x,y,z) dx dy dz\)
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\(a =\) \(b =\)
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\(g_1(z) =\) \(g_2(z) =\)
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\(h_1(y,z) =\) \(h_2(y,z) =\)
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4. \(\displaystyle \int_a^b \int_{g_1(y)}^{g_2(y)} \int_{h_1(y,z)}^{h_2(y,z)}f(x,y,z) dx dz dy\)
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\(a =\) \(b =\)
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\(g_1(y) =\) \(g_2(y) =\)
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\(h_1(y,z) =\) \(h_2(y,z) =\)
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5. \(\displaystyle \int_a^b \int_{g_1(x)}^{g_2(x)} \int_{h_1(x,z)}^{h_2(x,z)}f(x,y,z) dy dz dx\)
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\(a =\) \(b =\)
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\(g_1(x) =\) \(g_2(x) =\)
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\(h_1(x,z) =\) \(h_2(x,z) =\)
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6. \(\displaystyle \int_a^b \int_{g_1(z)}^{g_2(z)} \int_{h_1(x,z)}^{h_2(x,z)}f(x,y,z) dy dx dz\)
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\(a =\) \(b =\)
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\(g_1(z) =\) \(g_2(z) =\)
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\(h_1(x,z) =\) \(h_2(x,z) =\)
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9.

Calculate the volume under the elliptic paraboloid \(z = 3x^2 + 6y^2\) and over the rectangle \(R = [-3, 3] \times [-1, 1]\text{.}\)
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10.

The motion of a solid object can be analyzed by thinking of the mass as concentrated at a single point, the center of mass. If the object has density \(\rho(x,y,z)\) at the point \((x,y,z)\) and occupies a region \(W\text{,}\) then the coordinates \((\overline{x},\overline{y},\overline{z})\) of the center of mass are given by
\begin{equation*} \overline{x} = \frac{1}{m}\int_W x\rho \, dV \quad \overline{y} = \frac{1}{m}\int_W y\rho \, dV \quad \overline{z} = \frac{1}{m}\int_W z\rho \, dV, \end{equation*}
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Assume \(x\text{,}\) \(y\text{,}\) \(z\) are in cm. Let \(C\) be a solid cone with both height and radius 4 and contained between the surfaces \(z=\sqrt{x^2+y^2}\) and \(z=4\text{.}\) If \(C\) has constant mass density of 1 g/cm\(^3\text{,}\) find the \(z\)-coordinate of \(C\)’s center of mass.
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\(\overline z =\)
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(Include .)
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11.

Without calculation, decide if each of the integrals below are positive, negative, or zero. Let W be the solid bounded by \(z = \sqrt{x^2 + y^2}\) and \(z = 2\text{.}\)
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  1. \(\displaystyle \displaystyle \iiint\limits_W \left( z - 2 \right) \, dV\)
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  2. \(\displaystyle \displaystyle \iiint\limits_W \left( z - \sqrt{x^2+y^2} \right) \, dV\)
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  3. \(\displaystyle \displaystyle \iiint\limits_W e^{-xyz} \, dV\)
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12.

Set up a triple integral to find the mass of the solid tetrahedron bounded by the xy-plane, the yz-plane, the xz-plane, and the plane \(x/6 + y/5 + z/30 = 1\text{,}\) if the density function is given by \(\delta(x,y,z) = x + y\text{.}\) Write an iterated integral in the form below to find the mass of the solid.
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\(\displaystyle \iiint\limits_R f(x,y,z) \, dV = \int_A^B \!\! \int_C^D \!\! \int_E^F\) \(\, dz \, dy \, dx\)
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with limits of integration
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A =
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B =
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C =
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D =
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E =
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F =
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13.

Consider the solid \(S\) that is bounded by the parabolic cylinder \(y = x^2\) and the planes \(z=0\) and \(z=1-y\) as shown in Figure 12.6.11.
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described in detail following the image
The solid bounded by \(y = x^2\) and the planes \(z=0\) and \(z=1-y\text{.}\)
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Figure 12.6.11. The solid bounded by \(y = x^2\) and the planes \(z=0\) and \(z=1-y\text{.}\)
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Assume the density of \(S\) is given by \(\delta(x,y,z) = z\)
  1. Set up (but do not evaluate) an iterated integral that represents the mass of \(S\text{.}\) Integrate first with respect to \(z\text{,}\) then \(y\text{,}\) then \(x\text{.}\) A picture of the projection of \(S\) onto the \(xy\)-plane is shown at left in Figure 12.6.12.
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  2. Set up (but do not evaluate) an iterated integral that represents the mass of \(S\text{.}\) In this case, integrate first with respect to \(y\text{,}\) then \(z\text{,}\) then \(x\text{.}\) A picture of the projection of \(S\) onto the \(xz\)-plane is shown at center in Figure 12.6.12.
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  3. Set up (but do not evaluate) an iterated integral that represents the mass of \(S\text{.}\) For this integral, integrate first with respect to \(x\text{,}\) then \(y\text{,}\) then \(z\text{.}\) A picture of the projection of \(S\) onto the \(yz\)-plane is shown at right in Figure 12.6.12.
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  4. Which of these three orders of integration is the most natural to you? Why?
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described in detail following the image
Projections of \(S\) onto the \(xy\text{,}\) \(xz\text{,}\) and \(yz\)-planes.
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described in detail following the image
Projections of \(S\) onto the \(xy\text{,}\) \(xz\text{,}\) and \(yz\)-planes.
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described in detail following the image
Projections of \(S\) onto the \(xy\text{,}\) \(xz\text{,}\) and \(yz\)-planes.
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Figure 12.6.12. Projections of \(S\) onto the \(xy\text{,}\) \(xz\text{,}\) and \(yz\)-planes.
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14.

This problem asks you to investigate the average value of some different quantities.
  1. Set up, but do not evaluate, an iterated integral expression whose value is the average sum of all real numbers \(x\text{,}\) \(y\text{,}\) and \(z\) that have the following property: \(y\) is between 0 and 2, \(x\) is greater than or equal to 0 but cannot exceed \(2y\text{,}\) and \(z\) is greater than or equal to 0 but cannot exceed \(x+y\text{.}\)
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  2. Set up, but do not evaluate, an integral expression whose value represents the average value of \(f(x,y,z) = x + y + z\) over the solid region in the first octant bounded by the surface \(z = 4 - x - y^2\) and the coordinate planes \(x=0\text{,}\) \(y=0\text{,}\) \(z=0\text{.}\)
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  3. How are the quantities in (a) and (b) similar? How are they different?
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15.

Consider the solid that lies between the paraboloids \(z = g(x,y) = x^2 + y^2\) and \(z = f(x,y) = 8 - 3x^2 - 3y^2\text{.}\)
  1. By eliminating the variable \(z\text{,}\) determine the curve of intersection between the two paraboloids, and sketch this curve in the \(xy\)-plane.
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  2. Set up, but do not evaluate, an iterated integral expression whose value determines the mass of the solid, integrating first with respect to \(z\text{,}\) then \(y\text{,}\) then \(x\text{.}\) Assume the the solid’s density is given by \(\delta(x,y,z) = \frac{1}{x^2 + y^2 + z^2 + 1}\text{.}\)
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  3. Set up, but do not evaluate, iterated integral expressions whose values determine the mass of the solid using all possible remaining orders of integration. Use \(\delta(x,y,z) = \frac{1}{x^2 + y^2 + z^2 + 1}\) as the density of the solid.
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  4. Set up, but do not evaluate, iterated integral expressions whose values determine the center of mass of the solid. Again, assume the the solid’s density is given by \(\delta(x,y,z) = \frac{1}{x^2 + y^2 + z^2 + 1}\text{.}\)
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  5. Which coordinates of the center of mass can you determine without evaluating any integral expression? Why?
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16.

In each of the following problems, your task is to
  • (i).
    sketch, by hand, the region over which you integrate
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  • (ii).
    set up iterated integral expressions which, when evaluated, will determine the value sought
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  • (iii).
    use appropriate technology to evaluate each iterated integral expression you develop
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Note well: in some problems you may be able to use a double rather than a triple integral, and polar coordinates may be helpful in some cases.
  1. Consider the solid created by the region enclosed by the circular paraboloid \(z = 4 - x^2 - y^2\) over the region \(R\) in the \(xy\)-plane enclosed by \(y = -x\) and the circle \(x^2 + y^2 = 4\) in the first, second, and fourth quadrants. Determine the solid’s volume.
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  2. Consider the solid region that lies beneath the circular paraboloid \(z = 9 - x^2 - y^2\) over the triangular region between \(y = x\text{,}\) \(y = 2x\text{,}\) and \(y = 1\text{.}\) Assuming that the solid has its density at point \((x,y,z)\) given by \(\delta(x,y,z) = xyz + 1\text{,}\) measured in grams per cubic cm, determine the center of mass of the solid.
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  3. In a certain room in a house, the walls can be thought of as being formed by the lines \(y = 0\text{,}\) \(y = 12 + x/4\text{,}\) \(x = 0\text{,}\) and \(x = 12\text{,}\) where length is measured in feet. In addition, the ceiling of the room is vaulted and is determined by the plane \(z = 16 - x/6 - y/3\text{.}\) A heater is stationed in the corner of the room at \((0,0,0)\) and causes the temperature in the room at a particular time to be given by
    \begin{equation*} T(x,y,z) = \frac{80}{1 + \frac{x^2}{1000} + \frac{y^2}{1000} + \frac{z^2}{1000}} \end{equation*}
    What is the average temperature in the room?
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  4. Consider the solid enclosed by the cylinder \(x^2 + y^2 = 9\) and the planes \(y + z = 5\) and \(z = 1\text{.}\) Assuming that the solid’s density is given by \(\delta(x,y,z) = \sqrt{x^2 + y^2}\text{,}\) find the mass and center of mass of the solid.
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