Motivating Questions
-
How are a triple Riemann sum and the corresponding triple integral of a continuous function \(f = f(x,y,z)\) defined?
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What are two things the triple integral of a function can tell us?
Section 12.6 Triple Integrals
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 \([a,b]\text{.}\) Moreover, this double integral has natural interpretations and applications, and can be considered over non-rectangular regions, \(D\text{.}\) For instance, given a continuous function \(f\) over a region \(D\text{,}\) the average value of \(f\text{,}\) \(f_{\operatorname{AVG}(D)}\text{,}\) is given by
\begin{equation*}
f_{\operatorname{AVG}(D)} = \frac{1}{A(D)} \iint_D f(x,y) \, dA,
\end{equation*}
where \(A(D)\) is the area of \(D\text{.}\) Likewise, if \(\delta(x,y)\) describes the density function of a material over \(D\text{,}\) the total mass of material in \(D\text{,}\) \(M\text{,}\) is given by
\begin{equation*}
M = \iint_D \delta(x,y) \, dA.
\end{equation*}
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 a classical calculus approach for a rectangular box region in three-dimensions.
Preview Activity 12.6.1.
Consider a solid piece of granite in the shape of a box \(B = \{(x,y,z) : 0 \leq x \leq 4, 0 \leq y \leq 6, 0 \leq z \leq 8\}\text{,}\) whose density 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 (so we are measuring \(x\text{,}\) \(y\text{,}\) and \(z\) in meters). Our goal is to find the mass of this solid.
Recall that if the density was constant, we could find the mass by multiplying the density and volume; since the density varies from point to point, 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 is roughly constant and use these pieces to approximate the total volume.
(a)
For our approximation using smaller pieces, we will partition the interval \([0,4]\) into 2 subintervals of equal length, the interval \([0,6]\) into 3 subintervals of equal length, and the interval \([0,8]\) into 2 subintervals of equal length. This partitions the box \(B\) into sub-boxes as shown in Figure 12.6.1.
Let \(0=x_0 \lt x_1 \lt x_2=4\) be the endpoints of the subintervals of \([0,4]\) after partitioning. Draw a picture of Figure 12.6.1 by hand and label these endpoints on your drawing. Repeat this process with \(0=y_0 \lt y_1 \lt y_2 \lt y_3=6\) and \(0=z_0 \lt z_1 \lt z_2=8\text{.}\) What is the length \(\Delta x\) of each subinterval \([x_{i-1},x_i]\) for \(i\) from 1 to 2? the length of \(\Delta y\text{?}\) of \(\Delta z\text{?}\)
(b)
The partitions of the intervals \([0,4]\text{,}\) \([0,6]\) and \([0,8]\) partition the box \(B\) into sub-boxes. How many sub-boxes are there? What is volume \(\Delta V\) of each sub-box?
(c)
Let \(B_{ijk}\) denote the sub-box \([x_{i-1},x_i] \times [y_{j-1},y_j] \times [z_{k-1}, z_k]\text{.}\) Say that we choose a point \((x_{ijk}^*, y_{ijk}^*, z_{ijk}^*)\) in the \(i,j,k\)th sub-box for each possible combination of \(i,j,k\text{.}\) Write a couple of sentences to describe what physical quantity will \(\delta(x_{ijk}^*, y_{ijk}^*, z_{ijk}^*) \Delta V\) approximate and be sure to explain the meaning of \(\delta(x_{ijk}^*, y_{ijk}^*, z_{ijk}^*)\text{.}\)
(d)
We partitioned the space of inputs for \(\delta(x, y, z)\) into a few pieces. Write a few sentences about what you would change in the process above to improve the approximation of the total mass. Be sure to explain how you know that your changes will give a better approximation.
In the Preview Activity above, we covered 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.
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.
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 \(x_0\text{,}\) \(x_1\text{,}\) \(\ldots\text{,}\) \(x_m\) be the endpoints of these subintervals, where \(a = x_0\lt x_1\lt x_2 \lt \cdots \lt x_m = b\text{.}\) Do likewise with the interval \([c,d]\) using \(n\) subintervals of equal length \(\Delta y = \frac{d-c}{n}\) to generate \(c = y_0\lt y_1\lt y_2 \lt \cdots \lt y_n = d\text{,}\) and with the interval \([r,s]\) using \(\ell\) subintervals of equal length \(\Delta z = \frac{s-r}{\ell}\) to have \(r = z_0\lt z_1\lt z_2 \lt \cdots \lt z_l = s\text{.}\)
-
Let \(B_{ijk}\) be the sub-box of \(B\) with opposite vertices \((x_{i-1},y_{j-1},z_{k-1})\) and \((x_i, y_j, z_k)\) for \(i\) between \(1\) and \(m\text{,}\) \(j\) between \(1\) and \(n\text{,}\) and \(k\) between 1 and \(\ell\text{.}\) The volume of each \(B_{ijk}\) is \(\Delta V = \Delta x \cdot \Delta y \cdot \Delta z\text{.}\)
-
Let \((x_{ijk}^*, y_{ijk}^*, z_{ijk}^*)\) be a point in box \(B_{ijk}\) for each \(i\text{,}\) \(j\text{,}\) and \(k\text{.}\) 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. \end{equation*}
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); in this case, the finite sum of the mass approximations becomes the actual mass of the solid \(B\text{.}\) More generally, we have the following definition of the triple integral.
Definition 12.6.3.
With following notation defined as in 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.
\end{equation*}
As we noted 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, \(B\text{.}\)
Interpretations of Triple Integrals.
-
The triple integral\begin{equation*} \displaystyle V(S) = \iiint_S 1 \, dV \end{equation*}represents the volume of the solid \(S\).
-
The average value of the function \(f = f(x,y,x)\) over a solid domain \(S\) is given by\begin{equation*} f_{\operatorname{AVG}(S)} = \displaystyle \left(\frac{1}{V(S)} \right) \iiint_S f(x,y,z) \, dV, \end{equation*}where \(V(S)\) is the volume of the solid \(S\text{.}\)
-
The center of mass of the solid \(S\) with density \(\delta = \delta(x,y,z)\) is \((\overline{x}, \overline{y}, \overline{z})\text{,}\) where\begin{align*} \overline{x} \amp = \frac{\iiint_S x \ \delta(x,y,z) \, dV}{M},\\ \overline{y} \amp = \frac{\iiint_S y \ \delta(x,y,z) \, dV}{M}, \\ \overline{z} \amp = \frac{\iiint_S z \ \delta(x,y,z) \, dV}{M}, \end{align*}and \(M = \displaystyle \iiint_S \delta(x,y,z) \, dV\) is the mass of the solid \(S\text{.}\)
In a rectangular coordinate system, the volume element \(dV\) is \(dz \, dy \, dx\text{,}\) and, as a consequence, a triple integral of a function \(f\) over a box \(B = [a,b] \times [c,d] \times [r,s]\) in Cartesian 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*}
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.
Activity 12.6.2.
Set up and evaluate iterated integrals that will evaluate the triple integral of \(f(x,y,z) = x-y+2z\) over the box \(B = [-2,3] \times [1,4] \times [0,2]\text{.}\)
Activity 12.6.3.
Let \(S\) be the solid cone bounded by \(z = \sqrt{x^2+y^2}\) and \(z=3\text{.}\) A picture of \(S\) is shown in Figure 12.6.4. Our goal in 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{12.6.1}
\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 \((x,y,z)\text{.}\) Our particular task is to find the limits on each of the three integrals.
(a)
Remember that for the inner most integral of Equation (12.6.1), we will be looking for bounds on the \(z\)-coordinate for fixed values of \(x\) and \(y\text{.}\) In Figure 12.6.4, you can use the sliders to change the values of \(x\) and \(y\) and the plot will show the range of \(z\)-coordinates in our region for the particular values of \(x\) and \(y\text{.}\)
You should try several values of \(x\) and \(y\) and look at how the length of the segment changes in the \(z\)-direction. In particular, there is the same surface as the bottom boundary and the same surface as top boundary for every \(x\) and \(y\) pair. Give the equations for the top and bottom boundaries (in terms of \(x\) and \(y\))
\begin{align*}
z_{\text{top}}(x,y)\amp= \underline{\hspace{4cm}} \\
z_{\text{bottom}}(x,y)\amp= \underline{\hspace{4cm}}
\end{align*}
(b)
Now that we have our upper and lower boundaries for \(z\) as a function of a fixed choice of \(x\) and \(y\text{,}\) we need to look at what set of \((x,y)\) ,must be considered. Notice that if you choose a \((x,y)\) choice in Figure 12.6.4 that does not intersect our volume (like \((x,y)=(2.6,-2.3)\)), then the point shown is red. Write a couple of sentences to explain why the set of \((x,y)\) points we need to consider is NOT the square \([-3,3]\times[-3,3]\text{.}\)
(c)
On Figure 12.6.5, draw a plot of \(D\text{,}\) the region of \((x,y)\) points that you need to consider as part of \(S\text{.}\)
(d)
We now have part of our iterated integral
\begin{equation*}
\iiint_S \delta(x,y,z) dV = \int_D \left[ \int_{z_{\text{bottom}}(x,y)}^{z_{\text{top}}(x,y)} \delta(x,y,z) dz \right] dA
\end{equation*}
but we will need to give our region \(D\) from the previous task using a veritically simple description in order to have our iterated integral fit the form of Equation (12.6.1).
Give the inequalities that describe your region \(D\) from the previous task 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*}
(e)
In conclusion, write an iterated integral of the form (12.6.1) that represents the mass of the cone \(S\text{.}\)
Algebraic Note: When setting up iterated integrals, the limits on a given variable can be only in terms of the remaining variables. 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
-
\(\displaystyle \int_a^b \int_{g_1(x)}^{g_2(x)} \int_{h_1(x,y)}^{h_2(x,y)} f(x,y,z) \, dz \, dy \, dx\)
-
\(\displaystyle \int_r^s \int_{p_1(z)}^{p_2(z)} \int_{q_1(x,z)}^{q_2(x,z)} f(x,y,z) \, dy \, dx \, dz\)
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 (theoretically) be arranged in any order. Of course, 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.
Example 12.6.6.
In this example, we will find the mass of the tetrahedron 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 12.6.7.
We find the mass, \(M\text{,}\) of the tetrahedron by the triple integral
\begin{equation*}
M = \iiint_S \delta(x,y,z) \, dV,
\end{equation*}
where \(S\) is the solid tetrahedron described above. In this example, we choose to integrate with respect to \(z\) first for the innermost integral. The top of the tetrahedron is given by the equation
\begin{equation*}
x + 2 y + 3 z = 6;
\end{equation*}
solving for \(z\) then yields
\begin{equation*}
z = \frac{1}{3}(6 - x - 2y).
\end{equation*}
The bottom of the tetrahedron is the \(xy\)-plane, so the limits on \(z\) in the iterated integral will be \(0 \leq z \leq \frac{1}{3}(6-x-2y)\text{.}\)
To find the bounds on \(x\) and \(y\) we project the tetrahedron onto the \(xy\)-plane; this corresponds to setting \(z = 0\) in the equation \(z = \frac{1}{3}(6 - x - 2y)\text{.}\) The resulting relation between \(x\) and \(y\) is
\begin{equation*}
x + 2 y = 6.
\end{equation*}
If we choose to integrate with respect to \(y\) for the middle integral in the iterated integral, then the lower limit on \(y\) is the \(x\)-axis and the upper limit is the hypotenuse of the triangle. Note that the hypotenuse joins the points \((6,0)\) and \((0,3)\text{,}\) thus has equation \(y = 3 - \frac{1}{2}x\text{.}\) Thus, the bounds on \(y\) are \(0 \leq y \leq 3 - \frac{1}{2}x\text{.}\) Finally, the \(x\) values run from 0 to 6, so the iterated integral that gives the mass of the tetrahedron is
\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.\tag{12.6.2}
\end{equation}
Evaluating the 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*}
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.
Activity 12.6.4.
There are several other ways we could have set up the integral to give the mass of the tetrahedron in Example 12.6.6.
(a)
How many different orders of integration could be used for iterated integrals that are equal to the integral in Equation (12.6.2)?
(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 (12.6.2). Before you write down the integral, think about Figure 12.6.7, and draw an appropriate two-dimensional image of an important projection.
(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 (12.6.2). As in (b), think carefully about the geometry first and draw a plot of the appropriate projection.
(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 (12.6.2).
Now that we have begun to understand how to set up iterated triple integrals, we can apply them to determine important quantities, such as those found in the next activity.
Activity 12.6.5.
A solid \(S\) is bounded below by the square \(z=0\text{,}\) \(-1 \leq x \leq 1\text{,}\) \(-1 \leq y \leq 1\) and above by the surface \(z = 2-x^2-y^2\text{.}\) A picture of the solid is shown in Figure 12.6.9.
(a)
First, set up both 1) an iterated double integral to find the volume of the solid \(S\) as a double integral of a solid under a surface and 2)an iterated triple integral that gives the volume of the solid \(S\text{.}\) You do not need to evaluate either integral. Write a couple of sentences to compare these two approaches.
(b)
Set up (but do not evaluate) iterated integral expressions that will tell us the center of mass of \(S\text{,}\) if the density at point \((x,y,z)\) is \(\delta(x,y,z)=x^2+1\text{.}\)
(c)
Set up (but do not evaluate) an iterated integral to find the average density on \(S\) using the density function from part (b).
(d)
Use technology appropriately to evaluate the iterated integrals you determined in (a), (b), and (c); does the location you determined for the center of mass make sense?
Subsection 12.6.3 Summary
-
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 integral of \(f\) over \(B\) is defined as\begin{equation*} \iiint_B f(x,y,z) \, dV = \lim_{\Delta V \to 0} \sum_{i=1}^m \sum_{j=1}^n \sum_{k=1}^l f(x_{ijk}^*, y_{ijk}^*, z_{ijk}^*) \cdot \Delta V, \end{equation*}where the triple Riemann sum is defined in the usual way. The definition of the triple integral naturally extends to non-rectangular solid regions \(S\text{.}\)
-
The triple integral \(\iiint_S f(x,y,z) \, dV\) can tell us
-
the mass of the solid \(S\) if \(f\) represents the density of \(S\) at the point \((x,y,z)\text{.}\)
Moreover,\begin{equation*} f_{\operatorname{AVG}(S)} = \displaystyle \frac{1}{V(S)} \iiint_S f(x,y,z) \, dV, \end{equation*}is the average value of \(f\) over \(S\text{.}\)
Exercises 12.6.4 Exercises
1.
Find the triple integral of the function \(f(x,y,z) = x^{2}\cos\!\left(y+z\right)\) over the cube \(6 \leq x \leq 8\text{,}\) \(0 \leq y \leq \pi\text{,}\) \(0 \leq z \leq \pi.\)
2.
Evaluate the triple integral
\begin{equation*}
\int \!\! \int \!\! \int_{\mathbf{E}} xyz \, dV
\end{equation*}
where E is the solid: \(0 \leq z \leq 1 , \ \
0 \leq y \leq z, \ \ 0 \leq x \leq y\text{.}\)
3.
Find the mass of the rectangular prism \(0 \leq x \leq 1, \ \ 0
\leq y \leq 4, \ \
0 \leq z \leq 4\text{,}\) with density function \(\rho \left( x, y, z \right) = x\text{.}\)
4.
Find the average value of the function \(f \left( x, y, z \right) = y e^{-xy}\) over the rectangular prism \(0 \leq x \leq 2\text{,}\) \(0 \leq y \leq 3\text{,}\) \(0 \leq z \leq 2\)
5.
Find the volume of the solid bounded by the planes x = 0, y = 0, z = 0, and x + y + z = 4.
6.
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*}
Use these definitions to find the moment of inertia about the \(z\)-axis of the rectangular solid of mass \(27\) given by \(0 \le x \le 3\text{,}\) \(0 \le y \le 3\text{,}\) \(0 \le z \le 3\text{.}\)
\(I_x =\)
\(I_y =\)
\(I_z =\)
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 \(z =0, x = 0, z = y - 6 x\) and \(y = 12\text{.}\)
1. \(\displaystyle \int_a^b
\int_{g_1(x)}^{g_2(x)} \int_{h_1(x,y)}^{h_2(x,y)}f(x,y,z) dz dy dx\)
2. \(\displaystyle \int_a^b
\int_{g_1(y)}^{g_2(y)} \int_{h_1(x,y)}^{h_2(x,y)}f(x,y,z) dz dx dy\)
3. \(\displaystyle \int_a^b
\int_{g_1(z)}^{g_2(z)} \int_{h_1(y,z)}^{h_2(y,z)}f(x,y,z) dx dy dz\)
4. \(\displaystyle \int_a^b
\int_{g_1(y)}^{g_2(y)} \int_{h_1(y,z)}^{h_2(y,z)}f(x,y,z) dx dz dy\)
5. \(\displaystyle \int_a^b
\int_{g_1(x)}^{g_2(x)} \int_{h_1(x,z)}^{h_2(x,z)}f(x,y,z) dy dz dx\)
6. \(\displaystyle \int_a^b
\int_{g_1(z)}^{g_2(z)} \int_{h_1(x,z)}^{h_2(x,z)}f(x,y,z) dy dx dz\)
9.
Calculate the volume under the elliptic paraboloid \(z = 4x^2 + 8y^2\) and over the rectangle \(R = [-2, 2] \times [-3, 3]\text{.}\)
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*}
Assume \(x\text{,}\) \(y\text{,}\) \(z\) are in cm. Let \(C\) be a solid cone with both height and radius 5 and contained between the surfaces \(z=\sqrt{x^2+y^2}\) and \(z=5\text{.}\) If \(C\) has constant mass density of 3 g/cm\(^3\text{,}\) find the \(z\)-coordinate of \(C\)’s center of mass.
\(\overline z =\)
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 \(z = 2\text{.}\)
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 \(x/3 + y/2 + z/6 = 1\text{,}\) 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.
\(\displaystyle
\iiint\limits_R f(x,y,z) \, dV
=
\int_A^B \!\! \int_C^D \!\! \int_E^F\) \(\, dz \, dy \, dx\)
with limits of integration
A =
B =
C =
D =
E =
F =
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 12.6.10.
Assume the density of \(S\) is given by \(\delta(x,y,z) = z\)
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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 12.6.11.
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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 12.6.11.
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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 12.6.11.
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Which of these three orders of integration is the most natural to you? Why?
14.
This problem asks you to investigate the average value of some different quantities.
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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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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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How are the quantities in (a) and (b) similar? How are they different?
15.
Consider the solid that lies between the paraboloids \(z = g(x,y) = x^2 + y^2\) and \(z = f(x,y) = 8 - 3x^2 - 3y^2\text{.}\)
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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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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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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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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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Which coordinates of the center of mass can you determine without evaluating any integral expression? Why?
16.
In each of the following problems, your task is to
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(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
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.
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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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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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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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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.
