How To Set Up Triple Integrals

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Motivating Questions

• How are a triple Riemann sum and the corresponding triple integral of a continuous function \(f = f(x,y,z)\) divers?

• What are two things the triple integral of a function tin can tell us?

Nosotros accept at present learned that we ascertain 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, and that these ideas parallel the single-variable integral of a part \(grand = m(x)\) on an interval \([a,b]\text{.}\) Moreover, this double integral has natural interpretations and applications, and can even exist considered over non-rectangular regions, \(D\text{.}\) For case, given a continuous role \(f\) 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{one}{A(D)} \iint_D f(x,y) \, dA, \cease{equation*}

where \(A(D)\) is the area of \(D\text{.}\) Besides, if \(\delta(ten,y)\) describes a mass density function on a lamina over \(D\text{,}\) the mass, \(M\text{,}\) of the lamina is given by

\begin{equation*} Thou = \iint_D \delta(x,y) \, dA. \cease{equation*}

Information technology is natural to wonder if information technology is possible to extend these ideas of double Riemann sums and double integrals for functions of ii variables to triple Riemann sums and then triple integrals for functions of iii variables. We begin investigating in Preview Activity 11.7.i.

Preview Action eleven.7.ane .

Consider a solid piece of granite in the shape of a box \(B = \{(ten,y,z) : 0 \leq x \leq 4, 0 \leq y \leq 6, 0 \leq z \leq 8\}\text{,}\) whose density varies from point to point. Let \(\delta(x, y, z)\) represent the mass density of the piece of granite at indicate \((x,y,z)\) 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.

Call up that if the density was constant, we could discover the mass by multiplying the density and book; since the density varies from bespeak to point, we volition use the arroyo we did with two-variable lamina problems, and slice the solid into small pieces on which the density is roughly constant.

Partition the interval \([0,four]\) into 2 subintervals of equal length, the interval \([0,6]\) into iii subintervals of equal length, and the interval \([0,8]\) into ii subintervals of equal length. This partitions the box \(B\) into sub-boxes as shown in Figure xi.7.i.

Figure 11.7.1. A partitioned 3-dimensional domain.

1. Let \(0=x_0 \lt x_1 \lt x_2=4\) exist the endpoints of the subintervals of \([0,four]\) after partitioning. Draw a picture show of Effigy xi.7.1 and characterization these endpoints on your drawing. Do likewise with \(0=y_0 \lt y_1 \lt y_2 \lt y_3=6\) and \(0=z_0 \lt z_1 \lt z_2=8\) What is the length \(\Delta 10\) of each subinterval \([x_{i-ane},x_i]\) for \(i\) from ane to two? the length of \(\Delta y\text{?}\) of \(\Delta z\text{?}\)

2. The partitions of the intervals \([0,4]\text{,}\) \([0,half-dozen]\) and \([0,eight]\) partition the box \(B\) into sub-boxes. How many sub-boxes are in that location? What is volume \(\Delta V\) of each sub-box?

3. Let \(B_{ijk}\) denote the sub-box \([x_{i-i},x_i] \times [y_{j-i},y_j] \times [z_{g-1}, z_k]\text{.}\) Say that we choose a point \((x_{ijk}^*, y_{ijk}^*, z_{ijk}^*)\) in the \(i,j,k\)thursday sub-box for each possible combination of \(i,j,k\text{.}\) What is the meaning of \(\delta(x_{ijk}^*, y_{ijk}^*, z_{ijk}^*)\text{?}\) What physical quantity will \(\delta(x_{ijk}^*, y_{ijk}^*, z_{ijk}^*) \Delta V\) approximate?

4. What terminal stride(s) would it have to determine the exact mass of the piece of granite?

Subsection eleven.7.1 Triple Riemann Sums and Triple Integrals

Through the application of a mass density distribution over a iii-dimensional solid, Preview Activity 11.7.1 suggests that the generalization from double Riemann sums of functions of two variables to triple Riemann sums of functions of three variables is natural. In the same way, and then is the generalization from double integrals to triple integrals. By merely adding a \(z\)-coordinate to our earlier work, we tin can ascertain both a triple Riemann sum and the corresponding triple integral.

Definition eleven.7.two .

Allow \(f = f(ten,y,z)\) 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 equally follows.

• Partition the interval \([a,b]\) into \(g\) subintervals of equal length \(\Delta x = \frac{b-a}{m}\text{.}\) Allow \(x_0\text{,}\) \(x_1\text{,}\) \(\ldots\text{,}\) \(x_m\) be the endpoints of these subintervals, where \(a = x_0\lt x_1\lt x_2 \lt \cdots \lt x_m = b\text{.}\) Do besides with the interval \([c,d]\) using \(due north\) 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 \([r,due south]\) 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 \((x_{i-ane},y_{j-1},z_{k-1})\) and \((x_i, y_j, z_k)\) for \(i\) between \(1\) and \(m\text{,}\) \(j\) between \(1\) and \(n\text{,}\) and \(k\) between ane and \(\ell\text{.}\) The volume of each \(B_{ijk}\) is \(\Delta Five = \Delta x \cdot \Delta y \cdot \Delta z\text{.}\)

• Allow \((x_{ijk}^*, y_{ijk}^*, z_{ijk}^*)\) be a point in box \(B_{ijk}\) for each \(i\text{,}\) \(j\text{,}\) and \(one thousand\text{.}\) 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 the box \(B\text{,}\) so, as nosotros saw in Preview Activity 11.vii.i, the triple Riemann sum approximates the total mass of the box \(B\text{.}\) In order to find the exact mass of the box, we need to let the number of sub-boxes increment without bound (in other words, let \(m\text{,}\) \(due north\text{,}\) and \(\ell\) go to infinity); in this instance, the finite sum of the mass approximations becomes the actual mass of the solid \(B\text{.}\) More than by and large, we have the following definition of the triple integral.

Definition 11.7.3 .

With following notation defined equally in a triple Riemann sum, the triple integral of \(f\) over \(B\) is

\brainstorm{equation*} \iiint_B f(10,y,z) \, dV = \lim_{m,due north,\ell \to \infty} \sum_{i=one}^m \sum_{j=1}^n \sum_{g=ane}^{\ell} f(x_{ijk}^*, y_{ijk}^*, z_{ijk}^*) \cdot \Delta V. \end{equation*}

As nosotros noted earlier, if \(f(x, y, z)\) represents the density of the solid \(B\) at each signal \((10, y, z)\text{,}\) 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 role \(f\) over the solid \(B\text{,}\) we can use a triple integral to determine the average value of \(f\) over \(B\text{,}\) \(f_{\operatorname{AVG}(B)}\text{.}\) Nosotros note this generalization of our piece of work with functions of 2 variables forth 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 \(\R^three\text{,}\) not simply a box, \(B\text{.}\)

• The triple integral

  \begin{equation*} \displaystyle Five(S) = \iiint_S 1 \, dV \stop{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, \cease{equation*}

  where \(Five(S)\) is the book of the solid \(Southward\text{.}\)

• The center of mass of the solid \(Southward\) with density \(\delta = \delta(x,y,z)\) is \((\overline{10}, \overline{y}, \overline{z})\text{,}\) where

  \begin{align*} \overline{x} \amp = \frac{\iiint_S x \ \delta(x,y,z) \, dV}{1000},\\ \overline{y} \amp = \frac{\iiint_S y \ \delta(x,y,z) \, dV}{Grand}, \\ \overline{z} \amp = \frac{\iiint_S z \ \delta(x,y,z) \, dV}{Yard}, \end{marshal*}

  and \(Thousand = \displaystyle \iiint_S \delta(x,y,z) \, dV\) is the mass of the solid \(S\text{.}\)

In the Cartesian coordinate system, the book chemical element \(dV\) is \(dz \, dy \, dx\text{,}\) and, as a event, a triple integral of a function \(f\) over a box \(B = [a,b] \times [c,d] \times [r,southward]\) in Cartesian coordinates tin exist evaluated every bit an iterated integral of the form

\begin{equation*} \iiint_B f(ten,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 demand to describe the solid in terms of variable limits.

Activity 11.7.ii .

1. Set up and evaluate the triple integral of \(f(10,y,z) = x-y+2z\) over the box \(B = [-two,3] \times [1,4] \times [0,2]\text{.}\)

2. Allow \(South\) be the solid cone divisional by \(z = \sqrt{ten^two+y^2}\) and \(z=3\text{.}\) A flick of \(S\) is shown at right in Effigy 11.7.4. Our goal in what follows is to gear up an iterated integral of the form

   \begin{equation} \int_{ten=?}^{x=?} \int_{y=?}^{y=?} \int_{z=?}^{z=?} \delta(x,y,z) \, dz \, dy \, dx\label{eq_11_7_TI_not_box}\tag{11.seven.ane} \end{equation}

   to stand for the mass of \(Due south\) in the setting where \(\delta(x,y,z)\) tells us the density of \(S\) at the point \((x,y,z)\text{.}\) Our particular chore is to detect the limits on each of the three integrals.

   

   

   Figure xi.7.4. Left: The cone. Right: Its projection.

   1. If we recall about slicing up the solid, nosotros tin can consider slicing the domain of the solid's projection onto the \(xy\)-aeroplane (just every bit nosotros would slice a two-dimensional region in \(\R^2\)), and and so slice in the \(z\)-direction as well. The projection of the solid onto the \(xy\)-airplane is shown at left in Figure 11.7.4. If we decide to get-go piece the domain of the solid's projection perpendicular to the \(x\)-centrality, over what range of constant \(ten\)-values would nosotros take to slice?

   2. If nosotros continue with slicing the domain, what are the limits on \(y\) on a typical slice? How do these depend on \(x\text{?}\) What, therefore, are the limits on the middle integral?

   3. Finally, now that nosotros have thought virtually slicing up the two-dimensional domain that is the projection of the cone, what are the limits on \(z\) in the innermost integral? Note that over whatsoever point \((x,y)\) in the plane, a vertical piece in the \(z\) direction will involve a range of values from the cone itself to its flat top. In detail, observe that at least one of these limits is not constant but depends on \(ten\) and \(y\text{.}\)

   4. In decision, write an iterated integral of the form (11.vii.i) that represents the mass of the cone \(S\text{.}\)

Annotation well: When setting upwards iterated integrals, the limits on a given variable can be only in terms of the remaining variables. In add-on, in that location are multiple dissimilar ways we tin can choose to prepare such an integral. For case, two possibilities for iterated integrals that represent a triple integral \(\iiint_S f(x,y,z) \, dV\) over a solid \(S\) are

• \(\displaystyle \int_a^b \int_{g_1(x)}^{g_2(10)} \int_{h_1(x,y)}^{h_2(x,y)} f(x,y,z) \, dz \, dy \, dx\)

• \(\displaystyle \int_r^due south \int_{p_1(z)}^{p_2(z)} \int_{q_1(x,z)}^{q_2(x,z)} f(x,y,z) \, dy \, dx \, dz\)

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. In that location are 4 other options beyond the two stated hither, since the variables \(x\text{,}\) \(y\text{,}\) and \(z\) tin (theoretically) exist arranged in any gild. Of course, in many circumstances, an insightful selection of variable gild will go far easier to set upward an iterated integral, just as was the case when we worked with double integrals.

Example 11.7.5 .

Find the mass of the tetrahedron in the first octant bounded by the coordinate planes and the plane \(ten + 2 y + 3 z = six\) if the density at point \((x,y,z)\) is given by \(\delta(x, y, z) = 10 + y + z\text{.}\) A flick of the solid tetrahedron is shown at left in Effigy xi.7.half dozen.

Figure eleven.seven.6. Left: The tetrahedron. Correct: Its project.

We find the mass, \(Thousand\text{,}\) of the tetrahedron by the triple integral

\brainstorm{equation*} M = \iiint_S \delta(x,y,z) \, dV, \terminate{equation*}

where \(Due south\) is the solid tetrahedron described in a higher place. In this example, we choose to integrate with respect to \(z\) first for the innermost integral. The tiptop of the tetrahedron is given past the equation

\begin{equation*} ten + two y + 3 z = vi; \end{equation*}

solving for \(z\) then yields

\begin{equation*} z = \frac{1}{iii}(6 - x - 2y). \end{equation*}

The bottom of the tetrahedron is the \(xy\)-plane, so the limits on \(z\) in the iterated integral volition exist \(0 \leq z \leq \frac{ane}{iii}(vi-x-2y)\text{.}\)

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 betwixt \(x\) and \(y\) is

\begin{equation*} x + two y = half-dozen. \cease{equation*}

The correct image in Figure 11.7.6 shows the projection of the tetrahedron onto the \(xy\)-plane.

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 \(x\)-centrality and the upper limit is the hypotenuse of the triangle. Note that the hypotenuse joins the points \((six,0)\) and \((0,iii)\) and so has equation \(y = 3 - \frac{i}{2}10\text{.}\) Thus, the premises on \(y\) are \(0 \leq y \leq 3 - \frac{1}{two}x\text{.}\) Finally, the \(x\) values run from 0 to six, and so the iterated integral that gives the mass of the tetrahedron is

\brainstorm{equation} M = \int_{0}^{6} \int_{0}^{iii-(i/2)x} \int_{0}^{(one/three)(vi-x-2y)} x+y+z \, dz \, dy \, dx.\label{eq_11_7_Tetrahedron_mass}\tag{11.7.2} \finish{equation}

Evaluating the triple integral gives u.s.a.

\begin{align*} Thou \amp = \int_{0}^{6} \int_{0}^{3-(ane/2)x} \int_{0}^{(ane/three)(6-x-2y)} 10+y+z \, dz \, dy \, dx\\ \amp = \int_{0}^{half dozen} \int_{0}^{3-(1/2)x} \left[xz+yz+\frac{z}{2}\right]\biggm|_{0}^{(1/3)(half-dozen-x-2y)} \, dy \, dx\\ \amp = \int_{0}^{6} \int_{0}^{3-(1/2)x} \frac{4}{iii}x - \frac{5}{xviii}x^two - \frac{}{nine}xy + \frac{2}{3}y - \frac{four}{ix}y^2 + 2 \, dy \, dx\\ \amp = \int_{0}^{6} \left[\frac{4}{three}xy - \frac{5}{18}ten^2y - \frac{7}{eighteen}xy^2 + \frac{ane}{3}y^2 - \frac{4}{27}y^3 + 2y \right]\biggm|_{0}^{3-(1/2)x} \, dx\\ \amp = \int_{0}^{6} 5 + \frac{1}{2}ten - \frac{7}{12}x^two + \frac{13}{216}x^3 \, dx\\ \amp = \left[5x + \frac{1}{4}x^2 - \frac{vii}{36}ten^three + \frac{13}{864}10^4 \right] \biggm|_{0}^{half-dozen}\\ \amp = \frac{33}{2}. \end{align*}

Setting up limits on iterated integrals can require considerable geometric intuition. It is important to not only create advisedly labeled figures, only too to recall most how we wish to slice the solid. Further, note that when we say "we volition integrate beginning with respect to \(x\text{,}\)" past "first" we are referring to the innermost integral in the iterated integral. The next activeness explores several different ways we might ready upwardly the integral in the preceding instance.

Activity eleven.7.3 .

At that place are several other ways nosotros could have set upwardly the integral to give the mass of the tetrahedron in Example 11.vii.5.

1. How many different iterated integrals could be ready that are equal to the integral in Equation (11.7.2)?

2. Set up an iterated integral, integrating kickoff with respect to \(z\text{,}\) then \(10\text{,}\) then \(y\) that is equivalent to the integral in Equation (11.7.ii). Before you lot write down the integral, think about Figure 11.seven.half-dozen, and draw an advisable two-dimensional prototype of an important project.

3. Gear up an iterated integral, integrating first with respect to \(y\text{,}\) so \(z\text{,}\) so \(x\) that is equivalent to the integral in Equation (eleven.seven.2). Equally in (b), think advisedly about the geometry showtime.

4. Prepare up an iterated integral, integrating first with respect to \(x\text{,}\) then \(y\text{,}\) and so \(z\) that is equivalent to the integral in Equation (11.seven.2).

Now that we have begun to sympathize how to set up iterated triple integrals, we tin can apply them to determine of import quantities, such as those found in the next activity.

Action 11.7.4 .

A solid \(Southward\) is bounded below past the square \(z=0\text{,}\) \(-1 \leq ten \leq one\text{,}\) \(-1 \leq y \leq 1\) and above by the surface \(z = 2-ten^two-y^ii\text{.}\) A picture of the solid is shown in Figure xi.seven.7.

Figure 11.7.7. The solid bounded by the surface \(z = two-x^2-y^2\text{.}\)

1. Starting time, set upwardly an iterated double integral to find the book of the solid \(South\) as a double integral of a solid under a surface. Then fix up an iterated triple integral that gives the volume of the solid \(S\text{.}\) Y'all practise not need to evaluate either integral. Compare the 2 approaches.

2. Set upwardly (only do not evaluate) iterated integral expressions that volition tell us the center of mass of \(S\text{,}\) if the density at point \((x,y,z)\) is \(\delta(ten,y,z)=ten^2+1\text{.}\)

3. Set up up (merely exercise not evaluate) an iterated integral to find the average density on \(S\) using the density function from part (b).

4. Use technology appropriately to evaluate the iterated integrals you determined in (a), (b), and (c); does the location you adamant for the middle of mass make sense?

Subsection 11.vii.2 Summary

• Allow \(f = f(x,y,z)\) be a continuous function on a box \(B = [a,b] \times [c,d] \times [r,south]\text{.}\) The triple integral of \(f\) over \(B\) is defined as

  \begin{equation*} \iiint_B f(10,y,z) \, dV = \lim_{\Delta 5 \to 0} \sum_{i=1}^m \sum_{j=1}^n \sum_{k=ane}^fifty f(x_{ijk}^*, y_{ijk}^*, z_{ijk}^*) \cdot \Delta V, \end{equation*}

  where the triple Riemann sum is divers in the usual way. The definition of the triple integral naturally extends to not-rectangular solid regions \(S\text{.}\)

• The triple integral \(\iiint_S f(10,y,z) \, dV\) tin tell us

  • -.

    the volume of the solid \(Due south\) if \(f(x,y,z) = 1\text{,}\)

  • -.

    the mass of the solid \(Due south\) 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, \finish{equation*}

  is the average value of \(f\) over \(S\text{.}\)

Exercises 11.7.3 Exercises

i.

Notice the triple integral of the part \(f(x,y,z) = x^{ii}\cos\!\left(y+z\right)\) over the cube \(vi \leq x \leq 8\text{,}\) \(0 \leq y \leq \pi\text{,}\) \(0 \leq z \leq \pi.\)

2.

Evaluate the triple integral

\begin{equation*} \int \!\! \int \!\! \int_{\mathbf{Eastward}} xyz \, dV \end{equation*}

where E is the solid: \(0 \leq z \leq 1 , \ \ 0 \leq y \leq z, \ \ 0 \leq ten \leq y\text{.}\)

3.

Observe the mass of the rectangular prism \(0 \leq x \leq 1, \ \ 0 \leq y \leq four, \ \ 0 \leq z \leq iv\text{,}\) with density function \(\rho \left( 10, y, z \right) = x\text{.}\)

4.

Find the average value of the role \(f \left( x, y, z \right) = y east^{-xy}\) over the rectangular prism \(0 \leq 10 \leq two\text{,}\) \(0 \leq y \leq 3\text{,}\) \(0 \leq z \leq 2\)

five.

Find the volume of the solid bounded past the planes ten = 0, y = 0, z = 0, and 10 + y + z = 4.

vi.

Find the mass of the solid bounded by the \(xy\)-plane, \(yz\)-plane, \(xz\)-airplane, and the plane \((x/3)+(y/4)+(z/12) = 1\text{,}\) if the density of the solid is given by \(\delta (x,y,z)=x + ii y\text{.}\)

mass =

Reply.

\(\frac{three\cdot iii\cdot four\cdot 4\cdot \left(3+4\cdot 2\right)}{24}\)

vii.

The moment of inertia of a solid body about an axis in 3-space relates the angular acceleration well-nigh this axis to torque (strength 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 book \(V\) are defined to be

\begin{equation*} I_x = \frac{k}{V}\int_W (y^two+z^2) \,dV\qquad I_y = \frac{m}{V}\int_W (x^2+z^ii) \,dV\qquad I_z = \frac{m}{V}\int_W (ten^two+y^2) \,dV \end{equation*}

Utilize these definitions to find the moment of inertia near the \(z\)-axis of the rectangular solid of mass \(27\) given past \(0 \le x \le 3\text{,}\) \(0 \le y \le 3\text{,}\) \(0 \le z \le 3\text{.}\)

\(I_x =\)

\(I_y =\)

\(I_z =\)

Respond. 1

\(\frac{1\cdot 3\cdot three\cdot 3\cdot \left(3\cdot 3+3\cdot 3\correct)}{3}\)

Answer. two

\(\frac{ane\cdot iii\cdot three\cdot three\cdot \left(3\cdot 3+3\cdot three\correct)}{3}\)

Reply. 3

\(\frac{one\cdot 3\cdot 3\cdot three\cdot \left(3\cdot iii+iii\cdot 3\right)}{3}\)

8.

Limited the integral \(\displaystyle \iiint_E f(10,y,z) dV\) as an iterated integral in six different ways, where East is the solid bounded by \(z =0, ten = 0, z = y - 6 10\) and \(y = 12\text{.}\)

1. \(\displaystyle \int_a^b \int_{g_1(x)}^{g_2(x)} \int_{h_1(10,y)}^{h_2(10,y)}f(x,y,z) dz dy dx\)

\(a =\) \(b =\)

\(g_1(x) =\) \(g_2(ten) =\)

\(h_1(x,y) =\) \(h_2(x,y) =\)

ii. \(\displaystyle \int_a^b \int_{g_1(y)}^{g_2(y)} \int_{h_1(x,y)}^{h_2(x,y)}f(x,y,z) dz dx dy\)

\(a =\) \(b =\)

\(g_1(y) =\) \(g_2(y) =\)

\(h_1(x,y) =\) \(h_2(x,y) =\)

three. \(\displaystyle \int_a^b \int_{g_1(z)}^{g_2(z)} \int_{h_1(y,z)}^{h_2(y,z)}f(x,y,z) dx dy dz\)

\(a =\) \(b =\)

\(g_1(z) =\) \(g_2(z) =\)

\(h_1(y,z) =\) \(h_2(y,z) =\)

4. \(\displaystyle \int_a^b \int_{g_1(y)}^{g_2(y)} \int_{h_1(y,z)}^{h_2(y,z)}f(x,y,z) dx dz dy\)

\(a =\) \(b =\)

\(g_1(y) =\) \(g_2(y) =\)

\(h_1(y,z) =\) \(h_2(y,z) =\)

v. \(\displaystyle \int_a^b \int_{g_1(10)}^{g_2(x)} \int_{h_1(x,z)}^{h_2(x,z)}f(ten,y,z) dy dz dx\)

\(a =\) \(b =\)

\(g_1(10) =\) \(g_2(x) =\)

\(h_1(x,z) =\) \(h_2(x,z) =\)

6. \(\displaystyle \int_a^b \int_{g_1(z)}^{g_2(z)} \int_{h_1(x,z)}^{h_2(x,z)}f(ten,y,z) dy dx dz\)

\(a =\) \(b =\)

\(g_1(z) =\) \(g_2(z) =\)

\(h_1(x,z) =\) \(h_2(ten,z) =\)

9.

Calculate the volume under the elliptic paraboloid \(z = 4x^2 + 8y^2\) and over the rectangle \(R = [-2, ii] \times [-3, iii]\text{.}\)

10.

The move of a solid object tin can be analyzed by thinking of the mass every bit concentrated at a unmarried point, the center of mass. If the object has density \(\rho(x,y,z)\) at the point \((x,y,z)\) and occupies a region \(W\text{,}\) and then the coordinates \((\overline{x},\overline{y},\overline{z})\) of the center of mass are given past

\begin{equation*} \overline{ten} = \frac{1}{g}\int_W ten\rho \, dV \quad \overline{y} = \frac{1}{m}\int_W y\rho \, dV \quad \overline{z} = \frac{1}{grand}\int_W z\rho \, dV, \finish{equation*}

Assume \(x\text{,}\) \(y\text{,}\) \(z\) are in cm. Let \(C\) be a solid cone with both pinnacle and radius five and independent between the surfaces \(z=\sqrt{x^2+y^2}\) and \(z=five\text{.}\) If \(C\) has constant mass density of 3 thousand/cm\(^three\text{,}\) discover the \(z\)-coordinate of \(C\)'southward centre of mass.

\(\overline z =\)

(Include units.)

11.

Without adding, decide if each of the integrals below are positive, negative, or zippo. Let W exist the solid bounded past \(z = \sqrt{x^2 + y^2}\) and \(z = two\text{.}\)

1. \(\displaystyle \displaystyle \iiint\limits_W \left( z - \sqrt{10^ii+y^ii} \right) \, dV\)

2. \(\displaystyle \displaystyle \iiint\limits_W \left( z - ii \right) \, dV\)

3. \(\displaystyle \displaystyle \iiint\limits_W e^{-xyz} \, dV\)

12.

Gear up a triple integral to find the mass of the solid tetrahedron divisional by the xy-aeroplane, the yz-aeroplane, the xz-plane, and the plane \(x/3 + y/ii + z/six = ane\text{,}\) if the density role is given by \(\delta(x,y,z) = x + y\text{.}\) Write an iterated integral in the class beneath to find the mass of the solid.

\(\displaystyle \iiint\limits_R f(10,y,z) \, dV = \int_A^B \!\! \int_C^D \!\! \int_E^F\) \(\, dz \, dy \, dx\)

with limits of integration

A =

B =

C =

D =

E =

F =

13.

Consider the solid \(S\) that is bounded by the parabolic cylinder \(y = 10^2\) and the planes \(z=0\) and \(z=1-y\) as shown in Figure 11.7.8.

Figure 11.vii.eight. The solid bounded by \(y = x^ii\) and the planes \(z=0\) and \(z=1-y\text{.}\)

Assume the density of \(South\) is given by \(\delta(x,y,z) = z\)

1. Set up (but exercise not evaluate) an iterated integral that represents the mass of \(Due south\text{.}\) Integrate first with respect to \(z\text{,}\) and so \(y\text{,}\) then \(x\text{.}\) A moving picture of the projection of \(S\) onto the \(xy\)-plane is shown at left in Effigy 11.seven.nine.

2. Set upward (but practise not evaluate) an iterated integral that represents the mass of \(S\text{.}\) In this instance, integrate start with respect to \(y\text{,}\) and then \(z\text{,}\) then \(x\text{.}\) A moving-picture show of the projection of \(Southward\) onto the \(xz\)-plane is shown at eye in Effigy 11.vii.ix.

3. Ready (only do not evaluate) an iterated integral that represents the mass of \(S\text{.}\) For this integral, integrate commencement with respect to \(x\text{,}\) then \(y\text{,}\) then \(z\text{.}\) A moving picture of the projection of \(Due south\) onto the \(yz\)-plane is shown at right in Figure xi.seven.9.

4. Which of these three orders of integration is the most natural to you? Why?

Figure 11.7.9. Projections of \(S\) onto the \(xy\text{,}\) \(xz\text{,}\) and \(yz\)-planes.

14.

This problem asks yous to investigate the boilerplate value of some different quantities.

1. Gear up up, but practice not evaluate, an iterated integral expression whose value is the average sum of all real numbers \(x\text{,}\) \(y\text{,}\) and \(z\) that take the following holding: \(y\) 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 \(x+y\text{.}\)

2. Set upward, but exercise not evaluate, an integral expression whose value represents the average value of \(f(ten,y,z) = x + y + z\) over the solid region in the showtime octant bounded by the surface \(z = 4 - x - y^2\) and the coordinate planes \(x=0\text{,}\) \(y=0\text{,}\) \(z=0\text{.}\)

3. How are the quantities in (a) and (b) similar? How are they different?

15.

Consider the solid that lies between the paraboloids \(z = g(x,y) = x^two + y^2\) 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 bend in the \(xy\)-plane.

2. Ready 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 \(\delta(x,y,z) = \frac{1}{ten^2 + y^2 + z^2 + 1}\text{.}\)

3. Set, simply exercise non evaluate, iterated integral expressions whose values decide the mass of the solid using all possible remaining orders of integration. Use \(\delta(10,y,z) = \frac{1}{10^2 + y^2 + z^ii + 1}\) as the density of the solid.

4. Fix, but do non evaluate, iterated integral expressions whose values make up one's mind the eye of mass of the solid. Again, assume the the solid'due south density is given by \(\delta(x,y,z) = \frac{i}{ten^2 + y^2 + z^ii + one}\text{.}\)

5. Which coordinates of the eye of mass can you decide without evaluating whatsoever integral expression? Why?

sixteen.

In each of the following issues, your task is to

• (i).

  sketch, by hand, the region over which you integrate

• (two).

  set up iterated integral expressions which, when evaluated, will determine the value sought

• (iii).

  use appropriate technology to evaluate each iterated integral expression you develop

Notation well: in some issues you lot may be able to use a double rather than a triple integral, and polar coordinates may exist helpful in some cases.

1. Consider the solid created past the region enclosed past the circular paraboloid \(z = iv - x^2 - y^ii\) over the region \(R\) in the \(xy\)-aeroplane enclosed by \(y = -x\) and the circle \(x^two + y^2 = 4\) in the first, second, and fourth quadrants. Determine the solid'south volume.

2. Consider the solid region that lies below the round paraboloid \(z = 9 - x^two - y^two\) over the triangular region between \(y = x\text{,}\) \(y = 2x\text{,}\) and \(y = 1\text{.}\) Assuming that the solid has its density at point \((10,y,z)\) given by \(\delta(x,y,z) = xyz + 1\text{,}\) measured in grams per cubic cm, determine the center of mass of the solid.

3. In a certain room in a house, the walls can be thought of equally being formed by the lines \(y = 0\text{,}\) \(y = 12 + x/4\text{,}\) \(x = 0\text{,}\) and \(10 = 12\text{,}\) where length is measured in anxiety. In addition, the ceiling of the room is vaulted and is determined by the plane \(z = sixteen - x/six - y/3\text{.}\) A heater is stationed in the corner of the room at \((0,0,0)\) and causes the temperature in the room at a detail time to be given past

   \begin{equation*} T(x,y,z) = \frac{80}{1 + \frac{ten^2}{1000} + \frac{y^two}{yard} + \frac{z^2}{k}} \cease{equation*}

   What is the average temperature in the room?

4. Consider the solid enclosed by the cylinder \(x^ii + y^2 = 9\) and the planes \(y + z = five\) and \(z = 1\text{.}\) Bold that the solid'south density is given by \(\delta(x,y,z) = \sqrt{ten^2 + y^2}\text{,}\) find the mass and center of mass of the solid.

Source: https://activecalculus.org/multi/S-11-7-Triple-Integrals.html

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