4.7 triple integrals -...

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4.7. TRIPLE INTEGRALS 385 4.7 Triple Integrals 4.7.1 Introduction We dene triple integrals over a 3-D solid in a way similar to the denition of single integrals over an interval or double integrals over a 2-D region. We begin with the simplest case, when the region of integration is a rectangular box. We then extend the denition to more general bounded solids (3-D region), and focus on solids known as type 1, type2, and type 3 solids. 4.7.2 Triple Integrals Over a Rectangular Box Consider the rectangular box B = (x; y; z) 2 R 3 : a x b; c y d; r z s = [a; b] [c; d] [r; s] We proceed as follows: 1. Divide B into sub-boxes. To achieve this, we do the following: (a) We divide [a; b] into l subintervals [x i1 ;x i ] of equal width x. (b) We divide [c; d] into m subintervals [y j1 ;y j ] of equal width y. (c) We divide [r; s] into n subintervals [z k1 ;z k ] of equal width z. (d) Then, B is divided into lmn sub-boxes B ijk (see gure 4.29) where B ijk =[x i1 ;x i ] [y j1 ;y j ] [z k1 ;z k ] The volume of B ijk is V =xyz. 2. In each sub-box B ijk , we select a sample point x ijk ;y ijk ;z ijk . 3. We form the triple Riemann sum l X i=1 m X j=1 n X k=1 f x ijk ;y ijk ;z ijk V Denition 4.7.1 The triple integral of f over the box B is ZZZ B f (x; y; z) dV = lim l;m;n!1 l X i=1 m X j=1 n X k=1 f x ijk ;y ijk ;z ijk V It can be shown that the triple integral always exists if f is continuous. Computing such integrals will involve, as in the double integral case, iterated integrals. There is a version of Fubinis theorem for triple integrals.

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Page 1: 4.7 Triple Integrals - ksuweb.kennesaw.eduksuweb.kennesaw.edu/~plaval/math2203/multint_triple.pdfTRIPLE INTEGRALS 385 4.7 Triple Integrals ... It can be shown that the triple integral

4.7. TRIPLE INTEGRALS 385

4.7 Triple Integrals

4.7.1 Introduction

We define triple integrals over a 3-D solid in a way similar to the definition ofsingle integrals over an interval or double integrals over a 2-D region. We beginwith the simplest case, when the region of integration is a rectangular box. Wethen extend the definition to more general bounded solids (3-D region), andfocus on solids known as type 1, type2, and type 3 solids.

4.7.2 Triple Integrals Over a Rectangular Box

Consider the rectangular box

B ={

(x, y, z) ∈ R3 : a ≤ x ≤ b, c ≤ y ≤ d, r ≤ z ≤ s}

= [a, b]× [c, d]× [r, s]

We proceed as follows:

1. Divide B into sub-boxes. To achieve this, we do the following:

(a) We divide [a, b] into l subintervals [xi−1, xi] of equal width ∆x.

(b) We divide [c, d] into m subintervals [yj−1, yj ] of equal width ∆y.

(c) We divide [r, s] into n subintervals [zk−1, zk] of equal width ∆z.

(d) Then, B is divided into lmn sub-boxes Bijk (see figure 4.29) where

Bijk = [xi−1, xi]× [yj−1, yj ]× [zk−1, zk]

The volume of Bijk is ∆V = ∆x∆y∆z.

2. In each sub-box Bijk, we select a sample point(x∗ijk, y

∗ijk, z

∗ijk

).

3. We form the triple Riemann sum

l∑i=1

m∑j=1

n∑k=1

f(x∗ijk, y

∗ijk, z

∗ijk

)∆V

Definition 4.7.1 The triple integral of f over the box B is∫∫∫B

f (x, y, z) dV = liml,m,n→∞

l∑i=1

m∑j=1

n∑k=1

f(x∗ijk, y

∗ijk, z

∗ijk

)∆V

It can be shown that the triple integral always exists if f is continuous.

Computing such integrals will involve, as in the double integral case, iteratedintegrals. There is a version of Fubini’s theorem for triple integrals.

Page 2: 4.7 Triple Integrals - ksuweb.kennesaw.eduksuweb.kennesaw.edu/~plaval/math2203/multint_triple.pdfTRIPLE INTEGRALS 385 4.7 Triple Integrals ... It can be shown that the triple integral

386 CHAPTER 4. MULTIPLE INTEGRALS

Figure 4.29: Rectangular Box Divided in Sub-boxes

Page 3: 4.7 Triple Integrals - ksuweb.kennesaw.eduksuweb.kennesaw.edu/~plaval/math2203/multint_triple.pdfTRIPLE INTEGRALS 385 4.7 Triple Integrals ... It can be shown that the triple integral

4.7. TRIPLE INTEGRALS 387

Theorem 4.7.2 (Fubini’s Theorem for Triple Integrals) If f is continu-ous on the rectangular box B = [a, b]× [c, d]× [r, s] then∫∫∫

B

f (x, y, z) dV =

∫ b

a

∫ d

c

∫ s

r

f (x, y, z) dzdydx

Remark 4.7.3 Like in the double integral case, the order of integration can beswitched. There are six different iterated integrals which can be written. Weleave it to the reader to write these integrals.

Example 4.7.4 Evaluate∫∫∫

B

xyz2dV where B = [0, 1]× [−1, 2]× [0, 3].

By Fubini’s theorem, we have∫∫∫B

xyz2dV =

∫ 1

0

∫ 2

−1

∫ 3

0

xyz2dzdydx

The inner integrals is ∫ 3

0

xyz2dz =xyz3

3

∣∣∣∣30

= 9xy

Hence ∫ 2

−1

∫ 3

0

xyz2dzdy =

∫ 2

−19xydy

=9xy2

2

∣∣∣∣2−1

=36x

2− 9x

2

=27x

2

Finally ∫ 1

0

∫ 2

−1

∫ 3

0

xyz2dzdydx =

∫ 1

0

27x

2dx

=27x2

4

∣∣∣∣10

=27

4

Page 4: 4.7 Triple Integrals - ksuweb.kennesaw.eduksuweb.kennesaw.edu/~plaval/math2203/multint_triple.pdfTRIPLE INTEGRALS 385 4.7 Triple Integrals ... It can be shown that the triple integral

388 CHAPTER 4. MULTIPLE INTEGRALS

4.7.3 Triple Integrals Over a General Bounded 3-D Solids

We then proceed as we did for double integrals.

• We consider a general 3-D bounded solid E.

• Because E is bounded, we enclose it in a box B.

• We define a function F to agree with f on E and to be 0 on B − E.

• We define ∫∫∫E

f (x, y, z) dV =

∫∫∫B

F (x, y, z) dV

• We focus our attention to continuous functions f and to certain simpletypes of solids.

4.7.4 Triple Integrals Over a Type 1 Solid

A solid E is said to be of type 1 if it lies between the graphs of two continuousfunctions in x and y that is

E ={

(x, y, z) ∈ R3 : (x, y) ∈ D,u1 (x, y) ≤ z ≤ u2 (x, y)}

Where D is the projection of E onto the xy-plane. as shown in figure 4.30.Usingan argument similar to the one used for double integrals, it can be shown that∫∫∫

E

f (x, y, z) dV =

∫∫D

[∫ u2(x,y)

u1(x,y)

f (x, y, z) dz

]dA

To integrate the inner integral, we hold x and y as constants and integrate withrespect to z.If D is a type 1 region in the plane, as described in the previous section (see

figure 4.31), then

E ={

(x, y, z) ∈ R3 : a ≤ x ≤ b, g1 (x) ≤ y ≤ g2 (x) , u1 (x, y) ≤ z ≤ u2 (x, y)}

and ∫∫∫E

f (x, y, z) dV =

∫ b

a

∫ g2(x)

g1(x)

∫ u2(x,y)

u1(x,y)

f (x, y, z) dzdydx

If on the other hand If D is a type 2 region in the plane, as described in theprevious section (see figure 4.32), then

E ={

(x, y, z) ∈ R3 : c ≤ y ≤ d, h1 (y) ≤ x ≤ h2 (y) , u1 (x, y) ≤ z ≤ u2 (x, y)}

and ∫∫∫E

f (x, y, z) dV =

∫ d

c

∫ h2(y)

h1(y)

∫ u2(x,y)

u1(x,y)

f (x, y, z) dzdxdy

Page 5: 4.7 Triple Integrals - ksuweb.kennesaw.eduksuweb.kennesaw.edu/~plaval/math2203/multint_triple.pdfTRIPLE INTEGRALS 385 4.7 Triple Integrals ... It can be shown that the triple integral

4.7. TRIPLE INTEGRALS 389

Figure 4.30: A Type 1 Solid

Page 6: 4.7 Triple Integrals - ksuweb.kennesaw.eduksuweb.kennesaw.edu/~plaval/math2203/multint_triple.pdfTRIPLE INTEGRALS 385 4.7 Triple Integrals ... It can be shown that the triple integral

390 CHAPTER 4. MULTIPLE INTEGRALS

Figure 4.31: A Type 1 Solid

Page 7: 4.7 Triple Integrals - ksuweb.kennesaw.eduksuweb.kennesaw.edu/~plaval/math2203/multint_triple.pdfTRIPLE INTEGRALS 385 4.7 Triple Integrals ... It can be shown that the triple integral

4.7. TRIPLE INTEGRALS 391

Figure 4.32: A Type 1 Solid

Page 8: 4.7 Triple Integrals - ksuweb.kennesaw.eduksuweb.kennesaw.edu/~plaval/math2203/multint_triple.pdfTRIPLE INTEGRALS 385 4.7 Triple Integrals ... It can be shown that the triple integral

392 CHAPTER 4. MULTIPLE INTEGRALS

Figure 4.33: Solid E

Example 4.7.5 Evaluate∫∫∫

E

zdV where E is the solid tetrahedron bounded

by the four planes x = 0, y = 0, z = 0, and x+ y+ z = 1. Pictures of E and Dare shown on the next slide. The solid E and the region D are shown on figures4.33 and 4.34.Here, the solid E and the region D are shown. When working on a problem, itis wise to draw them. Looking at figure 4.33 we can see that u1 (x, y) = 0 andu2 (x, y) = 1− x− y. The region D can be viewed as a type 1 region that is

D ={

(x, y) ∈ R2 : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1− x}

Hence ∫∫∫E

zdV =

∫ 1

0

∫ 1−x

0

∫ 1−x−y

0

zdzdydx

First, we evaluate the innermost integral.∫ 1−x−y

0

zdz =z2

2

∣∣∣∣1−x−y0

=(1− x− y)

2

2

Page 9: 4.7 Triple Integrals - ksuweb.kennesaw.eduksuweb.kennesaw.edu/~plaval/math2203/multint_triple.pdfTRIPLE INTEGRALS 385 4.7 Triple Integrals ... It can be shown that the triple integral

4.7. TRIPLE INTEGRALS 393

Figure 4.34: Region D

Page 10: 4.7 Triple Integrals - ksuweb.kennesaw.eduksuweb.kennesaw.edu/~plaval/math2203/multint_triple.pdfTRIPLE INTEGRALS 385 4.7 Triple Integrals ... It can be shown that the triple integral

394 CHAPTER 4. MULTIPLE INTEGRALS

The next inner integral is then∫ 1−x

0

∫ 1−x−y

0

zdzdy =1

2

∫ 1−x

0

(1− x− y) dy

=1

2

− (1− x− y)3

3

∣∣∣∣∣1−x

0

=

1

6

(−0 + (1− x)

3)

Finally ∫∫∫E

zdV =1

6

∫ 1

0

(1− x)3dx

=1

6

− (1− x)4

4

∣∣∣∣∣1

0

=

1

24(−0 + 1)

=1

24

4.7.5 Triple Integrals Over a Type 2 Solid

E is said to be a type 2 solid if it lies between the graphs of two functions of yand z in other words

E ={

(x, y, z) ∈ R3 : (x, y) ∈ D,u1 (y, z) ≤ x ≤ u2 (y, z)}

where D is the projection of E on the yz-plane. Such a region is shown on figure4.35.It can be proven that∫∫∫

E

f (x, y, z) dV =

∫∫D

[∫ u2(y,z)

u1(y,z)

f (x, y, z) dx

]dA

4.7.6 Triple Integrals Over a Type 3 Solid

E is said to be a type 3 solid if it lies between the graphs of two functions of xand z in other words

E ={

(x, y, z) ∈ R3 : (x, z) ∈ D,u1 (x, z) ≤ y ≤ u2 (x, z)}

where D is the projection of E on the xz-plane. Such a region is shown on figure4.36.

Page 11: 4.7 Triple Integrals - ksuweb.kennesaw.eduksuweb.kennesaw.edu/~plaval/math2203/multint_triple.pdfTRIPLE INTEGRALS 385 4.7 Triple Integrals ... It can be shown that the triple integral

4.7. TRIPLE INTEGRALS 395

Figure 4.35: A Type 2 Solid

Page 12: 4.7 Triple Integrals - ksuweb.kennesaw.eduksuweb.kennesaw.edu/~plaval/math2203/multint_triple.pdfTRIPLE INTEGRALS 385 4.7 Triple Integrals ... It can be shown that the triple integral

396 CHAPTER 4. MULTIPLE INTEGRALS

Figure 4.36: A Type 3 Solid

It can be proven that∫∫∫E

f (x, y, z) dV =

∫∫D

[∫ u2(x,z)

u1(x,z)

f (x, y, z) dy

]dA

Page 13: 4.7 Triple Integrals - ksuweb.kennesaw.eduksuweb.kennesaw.edu/~plaval/math2203/multint_triple.pdfTRIPLE INTEGRALS 385 4.7 Triple Integrals ... It can be shown that the triple integral

Bibliography

[1] Joel Hass, Maurice D. Weir, and George B. Thomas, University calculus:Early transcendentals, Pearson Addison-Wesley, 2012.

[2] James Stewart, Calculus, Cengage Learning, 2011.

[3] Michael Sullivan and Kathleen Miranda, Calculus: Early transcendentals,Macmillan Higher Education, 2014.

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