Calculus Volume 3

5.4Triple Integrals

Calculus Volume 35.4 Triple Integrals

Learning Objectives

• 5.4.1 Recognize when a function of three variables is integrable over a rectangular box.
• 5.4.2 Evaluate a triple integral by expressing it as an iterated integral.
• 5.4.3 Recognize when a function of three variables is integrable over a closed and bounded region.
• 5.4.4 Simplify a calculation by changing the order of integration of a triple integral.
• 5.4.5 Calculate the average value of a function of three variables.

In Double Integrals over Rectangular Regions, we discussed the double integral of a function $f(x,y)f(x,y)$ of two variables over a rectangular region in the plane. In this section we define the triple integral of a function $f(x,y,z)f(x,y,z)$ of three variables over a rectangular solid box in space, $ℝ3.ℝ3.$ Later in this section we extend the definition to more general regions in $ℝ3.ℝ3.$

Integrable Functions of Three Variables

We can define a rectangular box $BB$ in $ℝ3ℝ3$ as $B={(x,y,z)|a≤x≤b,c≤y≤d,e≤z≤f}.B={(x,y,z)|a≤x≤b,c≤y≤d,e≤z≤f}.$ We follow a similar procedure to what we did in Double Integrals over Rectangular Regions. We divide the interval $[a,b][a,b]$ into $ll$ subintervals $[xi−1,xi][xi−1,xi]$ of equal length $Δx=xi−xi−1l,Δx=xi−xi−1l,$ divide the interval $[c,d][c,d]$ into $mm$ subintervals $[yi−1,yi][yi−1,yi]$ of equal length $Δy=yj−yj−1m,Δy=yj−yj−1m,$ and divide the interval $[e,f][e,f]$ into $nn$ subintervals $[zi−1,zi][zi−1,zi]$ of equal length $Δz=zk−zk−1n.Δz=zk−zk−1n.$ Then the rectangular box $BB$ is subdivided into $lmnlmn$ subboxes $Bijk=[xi−1,xi]×[yi−1,yi]×[zi−1,zi],Bijk=[xi−1,xi]×[yi−1,yi]×[zi−1,zi],$ as shown in Figure 5.40.

Figure 5.40 A rectangular box in $ℝ3ℝ3$ divided into subboxes by planes parallel to the coordinate planes.

For each $i,j,andk,i,j,andk,$ consider a sample point $(xijk*,yijk*,zijk*)(xijk*,yijk*,zijk*)$ in each sub-box $Bijk.Bijk.$ We see that its volume is $ΔV=ΔxΔyΔz.ΔV=ΔxΔyΔz.$ Form the triple Riemann sum

$∑i=1l∑j=1m∑k=1nf(xijk*,yijk*,zijk*)ΔxΔyΔz.∑i=1l∑j=1m∑k=1nf(xijk*,yijk*,zijk*)ΔxΔyΔz.$

We define the triple integral in terms of the limit of a triple Riemann sum, as we did for the double integral in terms of a double Riemann sum.

Definition

The triple integral of a function $f(x,y,z)f(x,y,z)$ over a rectangular box $BB$ is defined as

$liml,m,n→∞∑i=1l∑j=1m∑k=1nf(xijk*,yijk*,zijk*)ΔxΔyΔz=∭Bf(x,y,z)dVliml,m,n→∞∑i=1l∑j=1m∑k=1nf(xijk*,yijk*,zijk*)ΔxΔyΔz=∭Bf(x,y,z)dV$
(5.10)

if this limit exists.

When the triple integral exists on $B,B,$ the function $f(x,y,z)f(x,y,z)$ is said to be integrable on $B.B.$ Also, the triple integral exists if $f(x,y,z)f(x,y,z)$ is continuous on $B.B.$ Therefore, we will use continuous functions for our examples. However, continuity is sufficient but not necessary; in other words, $ff$ is bounded on $BB$ and continuous except possibly on the boundary of $B.B.$ The sample point $(xijk*,yijk*,zijk*)(xijk*,yijk*,zijk*)$ can be any point in the rectangular sub-box $BijkBijk$ and all the properties of a double integral apply to a triple integral. Just as the double integral has many practical applications, the triple integral also has many applications, which we discuss in later sections.

Now that we have developed the concept of the triple integral, we need to know how to compute it. Just as in the case of the double integral, we can have an iterated triple integral, and consequently, a version of Fubini’s thereom for triple integrals exists.

Theorem 5.9

Fubini’s Theorem for Triple Integrals

If $f(x,y,z)f(x,y,z)$ is continuous on a rectangular box $B=[a,b]×[c,d]×[e,f],B=[a,b]×[c,d]×[e,f],$ then

$∭Bf(x,y,z)dV=∫ef∫cd∫abf(x,y,z)dxdydz.∭Bf(x,y,z)dV=∫ef∫cd∫abf(x,y,z)dxdydz.$

This integral is also equal to any of the other five possible orderings for the iterated triple integral.

For $a,b,c,d,e,a,b,c,d,e,$ and $ff$ real numbers, the iterated triple integral can be expressed in six different orderings:

$∫ef∫cd∫abf(x,y,z)dxdydz=∫ef(∫cd(∫abf(x,y,z)dx)dy)dz=∫cd(∫ef(∫abf(x,y,z)dx)dz)dy=∫ab(∫ef(∫cdf(x,y,z)dy)dz)dx=∫ef(∫ab(∫cdf(x,y,z)dy)dx)dz=∫ce(∫ab(∫eff(x,y,z)dz)dx)dy=∫ab(∫ce(∫eff(x,y,z)dz)dy)dx.∫ef∫cd∫abf(x,y,z)dxdydz=∫ef(∫cd(∫abf(x,y,z)dx)dy)dz=∫cd(∫ef(∫abf(x,y,z)dx)dz)dy=∫ab(∫ef(∫cdf(x,y,z)dy)dz)dx=∫ef(∫ab(∫cdf(x,y,z)dy)dx)dz=∫ce(∫ab(∫eff(x,y,z)dz)dx)dy=∫ab(∫ce(∫eff(x,y,z)dz)dy)dx.$

For a rectangular box, the order of integration does not make any significant difference in the level of difficulty in computation. We compute triple integrals using Fubini’s Theorem rather than using the Riemann sum definition. We follow the order of integration in the same way as we did for double integrals (that is, from inside to outside).

Example 5.36

Evaluating a Triple Integral

Evaluate the triple integral $∫z=0z=1∫y=2y=4∫x=−1x=5(x+yz2)dxdydz.∫z=0z=1∫y=2y=4∫x=−1x=5(x+yz2)dxdydz.$

Example 5.37

Evaluating a Triple Integral

Evaluate the triple integral $∭Bx2yzdV∭Bx2yzdV$ where $B={(x,y,z)|−2≤x≤1,0≤y≤3,1≤z≤5}B={(x,y,z)|−2≤x≤1,0≤y≤3,1≤z≤5}$ as shown in the following figure.

Figure 5.41 Evaluating a triple integral over a given rectangular box.
Checkpoint 5.23

Evaluate the triple integral $∭BzsinxcosydV∭BzsinxcosydV$ where $B={(x,y,z)|0≤x≤π,3π2≤y≤2π,1≤z≤3}.B={(x,y,z)|0≤x≤π,3π2≤y≤2π,1≤z≤3}.$

Triple Integrals over a General Bounded Region

We now expand the definition of the triple integral to compute a triple integral over a more general bounded region $EE$ in $ℝ3.ℝ3.$ The general bounded regions we will consider are of three types. First, let $DD$ be the bounded region that is a projection of $EE$ onto the $xyxy$-plane. Suppose the region $EE$ in $ℝ3ℝ3$ has the form

$E={(x,y,z)|(x,y)∈D,u1(x,y)≤z≤u2(x,y)}.E={(x,y,z)|(x,y)∈D,u1(x,y)≤z≤u2(x,y)}.$

For two functions $z=u1(x,y)z=u1(x,y)$ and $z=u2(x,y),z=u2(x,y),$ such that $u1(x,y)≤u2(x,y)u1(x,y)≤u2(x,y)$ for all $(x,y)(x,y)$ in $DD$ as shown in the following figure.

Figure 5.42 We can describe region $EE$ as the space between $u1(x,y)u1(x,y)$ and $u2(x,y)u2(x,y)$ above the projection $DD$ of $EE$ onto the $xyxy$-plane.
Theorem 5.10

Triple Integral over a General Region

The triple integral of a continuous function $f(x,y,z)f(x,y,z)$ over a general three-dimensional region

$E={(x,y,z)|(x,y)∈D,u1(x,y)≤z≤u2(x,y)}E={(x,y,z)|(x,y)∈D,u1(x,y)≤z≤u2(x,y)}$

in $ℝ3,ℝ3,$ where $DD$ is the projection of $EE$ onto the $xyxy$-plane, is

$∭Ef(x,y,z)dV=∬D[∫u1(x,y)u2(x,y)f(x,y,z)dz]dA.∭Ef(x,y,z)dV=∬D[∫u1(x,y)u2(x,y)f(x,y,z)dz]dA.$

Similarly, we can consider a general bounded region $DD$ in the $xyxy$-plane and two functions $y=u1(x,z)y=u1(x,z)$ and $y=u2(x,z)y=u2(x,z)$ such that $u1(x,z)≤u2(x,z)u1(x,z)≤u2(x,z)$ for all $(x,z)(x,z)$ in $D.D.$ Then we can describe the solid region $EE$ in $ℝ3ℝ3$ as

$E={(x,y,z)|(x,z)∈D,u1(x,z)≤y≤u2(x,z)}E={(x,y,z)|(x,z)∈D,u1(x,z)≤y≤u2(x,z)}$

where $DD$ is the projection of $EE$ onto the $xyxy$-plane and the triple integral is

$∭Ef(x,y,z)dV=∬D[∫u1(x,z)u2(x,z)f(x,y,z)dy]dA.∭Ef(x,y,z)dV=∬D[∫u1(x,z)u2(x,z)f(x,y,z)dy]dA.$

Finally, if $DD$ is a general bounded region in the $yzyz$-plane and we have two functions $x=u1(y,z)x=u1(y,z)$ and $x=u2(y,z)x=u2(y,z)$ such that $u1(y,z)≤u2(y,z)u1(y,z)≤u2(y,z)$ for all $(y,z)(y,z)$ in $D,D,$ then the solid region $EE$ in $ℝ3ℝ3$ can be described as

$E={(x,y,z)|(y,z)∈D,u1(y,z)≤x≤u2(y,z)}E={(x,y,z)|(y,z)∈D,u1(y,z)≤x≤u2(y,z)}$

where $DD$ is the projection of $EE$ onto the $yzyz$-plane and the triple integral is

$∭Ef(x,y,z)dV=∬D[∫u1(y,z)u2(y,z)f(x,y,z)dx]dA.∭Ef(x,y,z)dV=∬D[∫u1(y,z)u2(y,z)f(x,y,z)dx]dA.$

Note that the region $DD$ in any of the planes may be of Type I or Type II as described in Double Integrals over General Regions. If $DD$ in the $xyxy$-plane is of Type I (Figure 5.43), then

$E={(x,y,z)|a≤x≤b,g1(x)≤y≤g2(x),u1(x,y)≤z≤u2(x,y)}.E={(x,y,z)|a≤x≤b,g1(x)≤y≤g2(x),u1(x,y)≤z≤u2(x,y)}.$
Figure 5.43 A box $EE$ where the projection $DD$ in the $xyxy$-plane is of Type I.

Then the triple integral becomes

$∭Ef(x,y,z)dV=∫ab∫g1(x)g2(x)∫u1(x,y)u2(x,y)f(x,y,z)dzdydx.∭Ef(x,y,z)dV=∫ab∫g1(x)g2(x)∫u1(x,y)u2(x,y)f(x,y,z)dzdydx.$

If $DD$ in the $xyxy$-plane is of Type II (Figure 5.44), then

$E={(x,y,z)|c≤x≤d,h1(x)≤y≤h2(x),u1(x,y)≤z≤u2(x,y)}.E={(x,y,z)|c≤x≤d,h1(x)≤y≤h2(x),u1(x,y)≤z≤u2(x,y)}.$
Figure 5.44 A box $EE$ where the projection $DD$ in the $xyxy$-plane is of Type II.

Then the triple integral becomes

$∭Ef(x,y,z)dV=∫y=cy=d∫x=h1(y)x=h2(y)∫z=u1(x,y)z=u2(x,y)f(x,y,z)dzdxdy.∭Ef(x,y,z)dV=∫y=cy=d∫x=h1(y)x=h2(y)∫z=u1(x,y)z=u2(x,y)f(x,y,z)dzdxdy.$

Example 5.38

Evaluating a Triple Integral over a General Bounded Region

Evaluate the triple integral of the function $f(x,y,z)=5x−3yf(x,y,z)=5x−3y$ over the solid tetrahedron bounded by the planes $x=0,y=0,z=0,x=0,y=0,z=0,$ and $x+y+z=1.x+y+z=1.$

Just as we used the double integral $∬D1dA∬D1dA$ to find the area of a general bounded region $D,D,$ we can use $∭E1dV∭E1dV$ to find the volume of a general solid bounded region $E.E.$ The next example illustrates the method.

Example 5.39

Finding a Volume by Evaluating a Triple Integral

Find the volume of a right pyramid that has the square base in the $xyxy$-plane $[−1,1]×[−1,1][−1,1]×[−1,1]$ and vertex at the point $(0,0,1)(0,0,1)$ as shown in the following figure.

Figure 5.46 Finding the volume of a pyramid with a square base.
Checkpoint 5.24

Consider the solid sphere $E={(x,y,z)|x2+y2+z2=9}.E={(x,y,z)|x2+y2+z2=9}.$ Write the triple integral $∭Ef(x,y,z)dV∭Ef(x,y,z)dV$ for an arbitrary function $ff$ as an iterated integral. Then evaluate this triple integral with $f(x,y,z)=1.f(x,y,z)=1.$ Notice that this gives the volume of a sphere using a triple integral.

Changing the Order of Integration

As we have already seen in double integrals over general bounded regions, changing the order of the integration is done quite often to simplify the computation. With a triple integral over a rectangular box, the order of integration does not change the level of difficulty of the calculation. However, with a triple integral over a general bounded region, choosing an appropriate order of integration can simplify the computation quite a bit. Sometimes making the change to polar coordinates can also be very helpful. We demonstrate two examples here.

Example 5.40

Changing the Order of Integration

Consider the iterated integral

$∫x=0x=1∫y=0y=x2∫z=0z=y2f(x,y,z)dzdydx.∫x=0x=1∫y=0y=x2∫z=0z=y2f(x,y,z)dzdydx.$

The order of integration here is first with respect to z, then y, and then x. Express this integral by changing the order of integration to be first with respect to x, then z, and then $y.y.$ Verify that the value of the integral is the same if we let $f(x,y,z)=xyz.f(x,y,z)=xyz.$

Checkpoint 5.25

Write five different iterated integrals equal to the given integral

$∫z=0z=4∫y=0y=4−z∫x=0x=yf(x,y,z)dxdydz.∫z=0z=4∫y=0y=4−z∫x=0x=yf(x,y,z)dxdydz.$

Example 5.41

Changing Integration Order and Coordinate Systems

Evaluate the triple integral $∭Ex2+z2dV,∭Ex2+z2dV,$ where $EE$ is the region bounded by the paraboloid $y=x2+z2y=x2+z2$ (Figure 5.48) and the plane $y=4.y=4.$

Figure 5.48 Integrating a triple integral over a paraboloid.

Average Value of a Function of Three Variables

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

Theorem 5.11

Average Value of a Function of Three Variables

If $f(x,y,z)f(x,y,z)$ is integrable over a solid bounded region $EE$ with positive volume $V(E),V(E),$ then the average value of the function is

$fave=1V(E)∭Ef(x,y,z)dV.fave=1V(E)∭Ef(x,y,z)dV.$

Note that the volume is $V(E)=∭E1dV.V(E)=∭E1dV.$

Example 5.42

Finding an Average Temperature

The temperature at a point $(x,y,z)(x,y,z)$ of a solid $EE$ bounded by the coordinate planes and the plane $x+y+z=1x+y+z=1$ is $T(x,y,z)=(xy+8z+20)°C.T(x,y,z)=(xy+8z+20)°C.$ Find the average temperature over the solid.

Checkpoint 5.26

Find the average value of the function $f(x,y,z)=xyzf(x,y,z)=xyz$ over the cube with sides of length $44$ units in the first octant with one vertex at the origin and edges parallel to the coordinate axes.

Section 5.4 Exercises

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

181.

$∭B(2x+3y2+4z3)dV,∭B(2x+3y2+4z3)dV,$ where $B={(x,y,z)|0≤x≤1,0≤y≤2,0≤z≤3}B={(x,y,z)|0≤x≤1,0≤y≤2,0≤z≤3}$

182.

$∭B(xy+yz+xz)dV,∭B(xy+yz+xz)dV,$ where $B={(x,y,z)|1≤x≤2,0≤y≤2,1≤z≤3}B={(x,y,z)|1≤x≤2,0≤y≤2,1≤z≤3}$

183.

$∭B(xcosy+z)dV,∭B(xcosy+z)dV,$ where $B={(x,y,z)|0≤x≤1,0≤y≤π,−1≤z≤1}B={(x,y,z)|0≤x≤1,0≤y≤π,−1≤z≤1}$

184.

$∭B(zsinx+y2)dV,∭B(zsinx+y2)dV,$ where $B={(x,y,z)|0≤x≤π,0≤y≤1,−1≤z≤2}B={(x,y,z)|0≤x≤π,0≤y≤1,−1≤z≤2}$

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

185.

$∫01∫12∫23(x2+lny+z)dxdydz∫01∫12∫23(x2+lny+z)dxdydz$

186.

$∫01∫−11∫03(zex+2y)dxdydz∫01∫−11∫03(zex+2y)dxdydz$

187.

$∫−12∫13∫04(x2z+1y)dxdydz∫−12∫13∫04(x2z+1y)dxdydz$

188.

$∫12∫−2−1∫01x+yzdxdydz∫12∫−2−1∫01x+yzdxdydz$

189.

Let $F,G,andHF,G,andH$ be continuous functions on $[a,b],[c,d],[a,b],[c,d],$ and $[e,f],[e,f],$ respectively, where $a,b,c,d,e,andfa,b,c,d,e,andf$ are real numbers such that $a Show that

$∫ab∫cd∫efF(x)G(y)H(z)dzdydx=(∫abF(x)dx)(∫cdG(y)dy)(∫efH(z)dz).∫ab∫cd∫efF(x)G(y)H(z)dzdydx=(∫abF(x)dx)(∫cdG(y)dy)(∫efH(z)dz).$
190.

Let $F,G,andHF,G,andH$ be differential functions on $[a,b],[c,d],[a,b],[c,d],$ and $[e,f],[e,f],$ respectively, where $a,b,c,d,e,andfa,b,c,d,e,andf$ are real numbers such that $a Show that

$∫ab∫cd∫efF′(x)G′(y)H′(z)dzdydx=[F(b)−F(a)][G(d)−G(c)][H(f)−H(e)].∫ab∫cd∫efF′(x)G′(y)H′(z)dzdydx=[F(b)−F(a)][G(d)−G(c)][H(f)−H(e)].$

In the following exercises, evaluate the triple integrals over the bounded region $E={(x,y,z)|a≤x≤b,h1(x)≤y≤h2(x),e≤z≤f}.E={(x,y,z)|a≤x≤b,h1(x)≤y≤h2(x),e≤z≤f}.$

191.

$∭E(2x+5y+7z)dV,∭E(2x+5y+7z)dV,$ where $E={(x,y,z)|0≤x≤1,0≤y≤−x+1,1≤z≤2}E={(x,y,z)|0≤x≤1,0≤y≤−x+1,1≤z≤2}$

192.

$∭E(ylnx+z)dV,∭E(ylnx+z)dV,$ where $E={(x,y,z)|1≤x≤e,0≤y≤lnx,0≤z≤1}E={(x,y,z)|1≤x≤e,0≤y≤lnx,0≤z≤1}$

193.

$∭E(sinx+siny)dV,∭E(sinx+siny)dV,$ where $E={(x,y,z)|0≤x≤π2,−cosx≤y≤cosx,−1≤z≤1}E={(x,y,z)|0≤x≤π2,−cosx≤y≤cosx,−1≤z≤1}$

194.

$∭E(xy+yz+xz)dV,∭E(xy+yz+xz)dV,$ where $E={(x,y,z)|0≤x≤1,−x2≤y≤x2,0≤z≤1}E={(x,y,z)|0≤x≤1,−x2≤y≤x2,0≤z≤1}$

In the following exercises, evaluate the triple integrals over the indicated bounded region $E.E.$

195.

$∭E(x+2yz)dV,∭E(x+2yz)dV,$ where $E={(x,y,z)|0≤x≤1,0≤y≤x,0≤z≤5−x−y}E={(x,y,z)|0≤x≤1,0≤y≤x,0≤z≤5−x−y}$

196.

$∭E(x3+y3+z3)dV,∭E(x3+y3+z3)dV,$ where $E={(x,y,z)|0≤x≤2,0≤y≤2x,0≤z≤4−x−y}E={(x,y,z)|0≤x≤2,0≤y≤2x,0≤z≤4−x−y}$

197.

$∭EydV,∭EydV,$ where $E={(x,y,z)|−1≤x≤1,−1−x2≤y≤1−x2,0≤z≤1−x2−y2}E={(x,y,z)|−1≤x≤1,−1−x2≤y≤1−x2,0≤z≤1−x2−y2}$

198.

$∭ExdV,∭ExdV,$ where $E={(x,y,z)|−2≤x≤2,−41−x2≤y≤4−x2,0≤z≤4−x2−y2}E={(x,y,z)|−2≤x≤2,−41−x2≤y≤4−x2,0≤z≤4−x2−y2}$

In the following exercises, evaluate the triple integrals over the bounded region $EE$ of the form $E={(x,y,z)|g1(y)≤x≤g2(y),c≤y≤d,e≤z≤f}.E={(x,y,z)|g1(y)≤x≤g2(y),c≤y≤d,e≤z≤f}.$

199.

$∭Ex2dV,∭Ex2dV,$ where $E={(x,y,z)|1−y2≤x≤y2−1,−1≤y≤1,1≤z≤2}E={(x,y,z)|1−y2≤x≤y2−1,−1≤y≤1,1≤z≤2}$

200.

$∭E(sinx+y)dV,∭E(sinx+y)dV,$ where $E={(x,y,z)|−y4≤x≤y4,0≤y≤2,0≤z≤4}E={(x,y,z)|−y4≤x≤y4,0≤y≤2,0≤z≤4}$

201.

$∭E(x−yz)dV,∭E(x−yz)dV,$ where $E={(x,y,z)|−y6≤x≤y,0≤y≤1,−1≤z≤1}E={(x,y,z)|−y6≤x≤y,0≤y≤1,−1≤z≤1}$

202.

$∭EzdV,∭EzdV,$ where $E={(x,y,z)|2−2y≤x≤2+y,0≤y≤1x,2≤z≤3}E={(x,y,z)|2−2y≤x≤2+y,0≤y≤1x,2≤z≤3}$

In the following exercises, evaluate the triple integrals over the bounded region

$E={(x,y,z)|g1(y)≤x≤g2(y),c≤y≤d,u1(x,y)≤z≤u2(x,y)}.E={(x,y,z)|g1(y)≤x≤g2(y),c≤y≤d,u1(x,y)≤z≤u2(x,y)}.$
203.

$∭EzdV,∭EzdV,$ where $E={(x,y,z)|−y≤x≤y,0≤y≤1,0≤z≤1−x4−y4}E={(x,y,z)|−y≤x≤y,0≤y≤1,0≤z≤1−x4−y4}$

204.

$∭E(xz+1)dV,∭E(xz+1)dV,$ where $E={(x,y,z)|0≤x≤y,0≤y≤2,0≤z≤1−x2−y2}E={(x,y,z)|0≤x≤y,0≤y≤2,0≤z≤1−x2−y2}$

205.

$∭E(x−z)dV,∭E(x−z)dV,$ where $E={(x,y,z)|−1−y2≤x≤0,0≤y≤12x,0≤z≤1−x2−y2}E={(x,y,z)|−1−y2≤x≤0,0≤y≤12x,0≤z≤1−x2−y2}$

206.

$∭E(x+y)dV,∭E(x+y)dV,$ where $E={(x,y,z)|0≤x≤1−y2,0≤y≤1,0≤z≤1−x}E={(x,y,z)|0≤x≤1−y2,0≤y≤1,0≤z≤1−x}$

In the following exercises, evaluate the triple integrals over the bounded region

$E={(x,y,z)|(x,y)∈D,u1(x,y)x≤z≤u2(x,y)},E={(x,y,z)|(x,y)∈D,u1(x,y)x≤z≤u2(x,y)},$ where $DD$ is the projection of $EE$ onto the $xyxy$-plane.

207.

$∬D(∫12(x+z)dz)dA,∬D(∫12(x+z)dz)dA,$ where $D={(x,y)|x2+y2≤1}D={(x,y)|x2+y2≤1}$

208.

$∬D(∫13x(z+1)dz)dA,∬D(∫13x(z+1)dz)dA,$ where $D={(x,y)|x2−y2≥1,x≤5}D={(x,y)|x2−y2≥1,x≤5}$

209.

$∬D(∫010−x−y(x+2z)dz)dA,∬D(∫010−x−y(x+2z)dz)dA,$ where $D={(x,y)|y≥0,x≥0,x+y≤10}D={(x,y)|y≥0,x≥0,x+y≤10}$

210.

$∬D(∫04x2+4y2ydz)dA,∬D(∫04x2+4y2ydz)dA,$ where $D={(x,y)|x2+y2≤4,y≥1,x≥0}D={(x,y)|x2+y2≤4,y≥1,x≥0}$

211.

The solid $EE$ bounded by $y2+z2=9,z=0,x=0,y2+z2=9,z=0,x=0,$ and $x=5x=5$ is shown in the following figure. Evaluate the integral $∭EzdV∭EzdV$ by integrating first with respect to $z,z,$ then $y,and thenx.y,and thenx.$

212.

The solid $EE$ bounded by $y=x,y=x,$ $x=4,x=4,$ $y=0,y=0,$ and $z=1z=1$ is given in the following figure. Evaluate the integral $∭ExyzdV∭ExyzdV$ by integrating first with respect to $x,x,$ then $y,y,$ and then $z.z.$

213.

[T] The volume of a solid $EE$ is given by the integral $∫−20∫x0∫0x2+y2dzdydx.∫−20∫x0∫0x2+y2dzdydx.$ Use a computer algebra system (CAS) to graph $EE$ and find its volume. Round your answer to two decimal places.

214.

[T] The volume of a solid $EE$ is given by the integral $∫−10∫−x20∫01+x2+y2dzdydx.∫−10∫−x20∫01+x2+y2dzdydx.$ Use a CAS to graph $EE$ and find its volume $V.V.$ Round your answer to two decimal places.

In the following exercises, use two circular permutations of the variables $x,y,andzx,y,andz$ to write new integrals whose values equal the value of the original integral. A circular permutation of $x,y,andzx,y,andz$ is the arrangement of the numbers in one of the following orders: $y,z,andxorz,x,andy.y,z,andxorz,x,andy.$

215.

$∫01∫13∫24(x2z2+1)dxdydz∫01∫13∫24(x2z2+1)dxdydz$

216.

$∫13∫01∫0−x+1(2x+5y+7z)dydxdz∫13∫01∫0−x+1(2x+5y+7z)dydxdz$

217.

$∫01∫−yy∫01−x4−y4lnxdzdxdy∫01∫−yy∫01−x4−y4lnxdzdxdy$

218.

$∫−11∫01∫−y6y(x+yz)dxdydz∫−11∫01∫−y6y(x+yz)dxdydz$

219.

Set up the integral that gives the volume of the solid $EE$ bounded by $y2=x2+z2y2=x2+z2$ and $y=a2,y=a2,$ where $a>0.a>0.$

220.

Set up the integral that gives the volume of the solid $EE$ bounded by $x=y2+z2x=y2+z2$ and $x=a2,x=a2,$ where $a>0.a>0.$

221.

Find the average value of the function $f(x,y,z)=x+y+zf(x,y,z)=x+y+z$ over the parallelepiped determined by $x=0,x=1,y=0,y=3,z=0,x=0,x=1,y=0,y=3,z=0,$ and $z=5.z=5.$

222.

Find the average value of the function $f(x,y,z)=xyzf(x,y,z)=xyz$ over the solid $E=[0,1]×[0,1]×[0,1]E=[0,1]×[0,1]×[0,1]$ situated in the first octant.

223.

Find the volume of the solid $EE$ that lies under the plane $x+y+z=9x+y+z=9$ and whose projection onto the $xyxy$-plane is bounded by $x=y−1,x=0,x=y−1,x=0,$ and $x+y=7.x+y=7.$

224.

Find the volume of the solid E that lies under the plane $2x+y+z=82x+y+z=8$ and whose projection onto the $xyxy$-plane is bounded by $x=sin−1y,y=0,x=sin−1y,y=0,$ and $x=π2.x=π2.$

225.

Consider the pyramid with the base in the $xyxy$-plane of $[−2,2]×[−2,2][−2,2]×[−2,2]$ and the vertex at the point $(0,0,8).(0,0,8).$

1. Show that the equations of the planes of the lateral faces of the pyramid are $4y+z=8,4y+z=8,$ $4y−z=−8,4y−z=−8,$ $4x+z=8,4x+z=8,$ and $−4x+z=8.−4x+z=8.$
2. Find the volume of the pyramid.
226.

Consider the pyramid with the base in the $xyxy$-plane of $[−3,3]×[−3,3][−3,3]×[−3,3]$ and the vertex at the point $(0,0,9).(0,0,9).$

1. Show that the equations of the planes of the side faces of the pyramid are
2. Find the volume of the pyramid.
227.

The solid $EE$ bounded by the sphere of equation $x2+y2+z2=r2x2+y2+z2=r2$ with $r>0r>0$ and located in the first octant is represented in the following figure.

1. Write the triple integral that gives the volume of $EE$ by integrating first with respect to $z,z,$ then with $y,y,$ and then with $x.x.$
2. Rewrite the integral in part a. as an equivalent integral in five other orders.
228.

The solid $EE$ bounded by the equation $9x2+4y2+z2=19x2+4y2+z2=1$ and located in the first octant is represented in the following figure.

1. Write the triple integral that gives the volume of $EE$ by integrating first with respect to $z,z,$ then with $y,y,$ and then with $x.x.$
2. Rewrite the integral in part a. as an equivalent integral in five other orders.
229.

Find the volume of the prism with vertices $(0,0,0),(2,0,0),(2,3,0),(0,0,0),(2,0,0),(2,3,0),$ $(0,3,0),(0,0,1),and(2,0,1).(0,3,0),(0,0,1),and(2,0,1).$

230.

Find the volume of the prism with vertices $(0,0,0),(4,0,0),(4,6,0),(0,0,0),(4,0,0),(4,6,0),$ $(0,6,0),(0,0,1),and(4,0,1).(0,6,0),(0,0,1),and(4,0,1).$

231.

The solid $EE$ bounded by $z=10−2x−yz=10−2x−y$ and situated in the first octant is given in the following figure. Find the volume of the solid.

232.

The solid $EE$ bounded by $z=1−x2z=1−x2$ and situated in the first octant is given in the following figure. Find the volume of the solid.

233.

The midpoint rule for the triple integral $∭Bf(x,y,z)dV∭Bf(x,y,z)dV$ over the rectangular solid box $BB$ is a generalization of the midpoint rule for double integrals. The region $BB$ is divided into subboxes of equal sizes and the integral is approximated by the triple Riemann sum $∑i=1l∑j=1m∑k=1nf(xi–,yj–,zk–)ΔV,∑i=1l∑j=1m∑k=1nf(xi–,yj–,zk–)ΔV,$ where $(xi–,yj–,zk–)(xi–,yj–,zk–)$ is the center of the box $BijkBijk$ and $ΔVΔV$ is the volume of each subbox. Apply the midpoint rule to approximate $∭Bx2dV∭Bx2dV$ over the solid $B={(x,y,z)|0≤x≤1,0≤y≤1,0≤z≤1}B={(x,y,z)|0≤x≤1,0≤y≤1,0≤z≤1}$ by using a partition of eight cubes of equal size. Round your answer to three decimal places.

234.

[T]

1. Apply the midpoint rule to approximate $∭Be−x2dV∭Be−x2dV$ over the solid $B={(x,y,z)|0≤x≤1,0≤y≤1,0≤z≤1}B={(x,y,z)|0≤x≤1,0≤y≤1,0≤z≤1}$ by using a partition of eight cubes of equal size. Round your answer to three decimal places.
2. Use a CAS to improve the above integral approximation in the case of a partition of $n3n3$ cubes of equal size, where $n=3,4,…,10.n=3,4,…,10.$
235.

Suppose that the temperature in degrees Celsius at a point $(x,y,z)(x,y,z)$ of a solid $EE$ bounded by the coordinate planes and $x+y+z=5x+y+z=5$ is $T(x,y,z)=xz+5z+10.T(x,y,z)=xz+5z+10.$ Find the average temperature over the solid.

236.

Suppose that the temperature in degrees Fahrenheit at a point $(x,y,z)(x,y,z)$ of a solid $EE$ bounded by the coordinate planes and $x+y+z=5x+y+z=5$ is $T(x,y,z)=x+y+xy.T(x,y,z)=x+y+xy.$ Find the average temperature over the solid.

237.

Show that the volume of a right square pyramid of height $hh$ and side length $aa$ is $v=ha23v=ha23$ by using triple integrals.

238.

Show that the volume of a regular right hexagonal prism of edge length $aa$ is $3a3323a332$ by using triple integrals.

239.

Show that the volume of a regular right hexagonal pyramid of edge length $aa$ is $a332a332$ by using triple integrals.

240.

If the charge density at an arbitrary point $(x,y,z)(x,y,z)$ of a solid $EE$ is given by the function $ρ(x,y,z),ρ(x,y,z),$ then the total charge inside the solid is defined as the triple integral $∭Eρ(x,y,z)dV.∭Eρ(x,y,z)dV.$ Assume that the charge density of the solid $EE$ enclosed by the paraboloids $x=5−y2−z2x=5−y2−z2$ and $x=y2+z2−5x=y2+z2−5$ is equal to the distance from an arbitrary point of $EE$ to the origin. Set up the integral that gives the total charge inside the solid $E.E.$

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