Calculus- Early Transcendentals, 2021a

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504 Multiple Integration Example 14.10: Volume of a Box Compute the volume of the box with opposite corners at (0,0,0) and (1,2,3). Solution. We use an integral to compute the volume of the box: ∫ 1 ∫ 2 ∫ 3 0 0 0 dzdydx = ∫ 1 ∫ 2 0 0 z| 3 0 dydx = ∫ 1 ∫ 2 0 0 3dydx = ∫ 1 0 ∫ 1 3y| 2 0 dx = 6dx = 6. 0 ♣ Of course, this is more interesting and useful when the limits are not constant. Example 14.11: Volume of a Tetrahedron Find the volume of the tetrahedron with corners at (0,0,0), (0,3,0), (2,3,0),and(2,3,5). Solution. The whole problem comes down to correctly describing the region by inequalities: 0 ≤ x ≤ 2, 3x/2 ≤ y ≤ 3, 0 ≤ z ≤ 5x/2. The lower y limit comes from the equation of the line y = 3x/2 that forms one edge of the tetrahedron in the xy-plane; the upper z limit comes from the equation of the plane z = 5x/2 that forms the “upper” side of the tetrahedron; see Figure 14.12. Now the volume is ∫ 2 ∫ 3 0 3x/2 ∫ 5x/2 0 dzdydx = = = ∫ 2 ∫ 3 0 3x/2 ∫ 2 ∫ 3 0 ∫ 2 0 ∫ 2 3x/2 z| 5x/2 0 dydx 5x 2 dydx ∣ 5x ∣∣∣ 3 2 y dx 3x/2 15x 2 − 15x2 = 0 4 dx = 15x2 4 − 15x3 2 12 ∣ 0 = 15 − 10 = 5. ♣
14.5. Triple Integrals 505 Figure 14.12: A tetrahedron. Pretty much just the way we did for two dimensions we can use triple integration to compute mass, center of mass, and various average quantities. Example 14.12: Average Temperature in a Cube Suppose the temperature at a point is given by T = xyz. Find the average temperature in the cube with opposite corners at (0,0,0) and (2,2,2). Solution. In two dimensions we add up the temperature at “each” point and divide by the area; here we add up the temperatures and divide by the volume, 8: ∫ 1 2 ∫ 2 ∫ 2 xyzdzdydx = 1 ∫ 2 ∫ 2 xyz 2 2 8 0 0 0 8 0 0 2 ∣ dydx = 1 ∫ 2 ∫ 2 xydydx 0 4 0 0 = 1 ∫ 2 xy 2 2 4 2 ∣ dx = 1 ∫ 2 xdx= 1 x 2 2 2 2 2 ∣ = 1. 0 0 0 0 ♣ Example 14.13: Mass & Center of Mass of Tetrahedron Suppose the density of an object is given by xz, and the object occupies the tetrahedron with corners (0,0,0), (0,1,0), (1,1,0),and(0,1,1). Find the mass and center of mass of the object. Solution. As usual, the mass is the integral of density over the region: M = ∫ 1 ∫ 1 ∫ y−x 0 x 0 xzdzdydx = ∫ 1 ∫ 1 0 x x(y − x) 2 dydx = 1 2 2 ∫ 1 0 x(1 − x) 3 dx 3
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with Open Texts Calculus Early Tran
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Calculus: Early Transcendentals an
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Calculus: Early Transcendentals an
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Table of Contents Table of Contents
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v 5.4.3 Taylor Polynomials . . . .
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vii 13.2 Limits and Continuity . .
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Introduction The emphasis in this c
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4 Review In the table, the set of r
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6 Review 1.1.3 The Quadratic Formul
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8 Review 1.1.4 Inequalities, Interv
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10 Review Inequality Rules Add/subt
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12 Review Guidelines for Solving Ra
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14 Review Looking where the
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16 Review Example 1.17: Absolute Va
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18 Review (a) |x|≥2 (b) |x − 3|
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20 Review The most familiar form of
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22 Review Example 1.25: Equa
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24 Review |Δy|, as shown in figure
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26 Review • (h,k) is the vert
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28 Review Determining the Type of C
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30 Review To determine a, we substi
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32 Review (a) A(2,0),B(4,3) (b) A(
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34 Review • Secant (abbreviated b
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36 Review Reading from the unit cir
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38 Review Notice that we can now fi
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40 Review 1.3.4 Graphs of Trigonome
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42 Review Exercises for 1.3 Exercis
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44 Review Exercise 1.4.7 Simplify t
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46 Functions Example 2.2: Domain of
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48 Functions Example 2.5: Domain Fi
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50 Functions For horizontal and ver
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52 Functions Solution. The domain o
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54 Functions After the positive int
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56 Functions Example 2.11: Determin
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58 Functions Example 2.16: Finding
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60 Functions Since the function f (
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62 Functions Example 2.21: Solve Lo
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64 Functions We can do something si
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66 Functions Cancellation Rules sin
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68 Functions (a) sin −1 ( √ 3/2
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70 Functions Example 2.32: Computin
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72 Functions (a) f (x) (b) − f (x
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Chapter 3 Limits 3.1 The Limit The
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3.2. Precise Definition of a Limit
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3.2. Precise Definition of a Limit
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3.3. Computing Limits: Graphically
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3.4. Computing Limits: Algebraicall
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3.4. Computing Limits: Algebraicall
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3.5. Infinite Limits and Limits at
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5 + x −1 (n) lim x→∞ 1 + 2x
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3.6. A Trigonometric Limit 99 Figur
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3.6. A Trigonometric Limit 101 Solu
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3.7. Continuity 103 § § ¥
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3.7. Continuity 105 Definition 3.43
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3.7. Continuity 107 Definition 3.45
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3.7. Continuity 109 Solution. We wi
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3.7. Continuity 111 Therefore, our
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3.7. Continuity 113 Exercise 3.7.3
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116 Derivatives doesn’t meet the
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118 Derivatives of the slope of the
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120 Derivatives algebra to find a s
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122 Derivatives we often use f and
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124 Derivatives you are required to
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126 Derivatives We can summarize
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128 Derivatives 4.2.3 Velocities Su
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130 Derivatives Exercise 4.2.5 Find
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132 Derivatives Proof. For convenie
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134 Derivatives Exercises for Secti
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136 Derivatives since sinx cosx −
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138 Derivatives In practice, of cou
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140 Derivatives Exercises for Secti
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142 Derivatives Exercise 4.5.39 Fin
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144 Derivatives . . . . Figure 4.4:
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146 Derivatives = ( d dx x2 ln2) e
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148 Derivatives Example 4.38: Deriv
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150 Derivatives Example 4.42: Equat
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152 Derivatives Solution. We take l
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154 Derivatives Exercise 4.7.8 Find
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156 Derivatives 4.8.1 Derivatives o
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158 Derivatives Exercise 4.8.4 The
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Chapter 5 Applications of Derivativ
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5.1. Related Rates 163 Solution. He
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5.1. Related Rates 165 Example 5.6:
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5.2. Extrema of a Function 167 Exer
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5.2. Extrema of a Function 169 poin
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5.2. Extrema of a Function 171 Exer
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5.2. Extrema of a Function 173 3. E
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5.2. Extrema of a Function 175 A si
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5.3. The Mean Value Theorem 177 2.
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5.3. The Mean Value Theorem 179
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5.3. The Mean Value Theorem 181 Exe
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5.4. Linear and Higher Order Approx
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5.5. L’Hôpital’s Rule 191 "bou
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5.5. L’Hôpital’s Rule 193 Exam
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5.5. L’Hôpital’s Rule 195 Exer
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5.6. Curve Sketching 197 consistent
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5.6. Curve Sketching 199 Solution.
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5.6. Curve Sketching 201 the concav
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5.6. Curve Sketching 203 If there a
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5.6. Curve Sketching 205 Note that
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5.7. Optimization Problems 207 Guid
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5.7. Optimization Problems 209 is t
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5.7. Optimization Problems 211 (Alt
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5.7. Optimization Problems 213 Exer
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Chapter 6 Integration 6.1 Displacem
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6.1. Displacement and Area 217 Exam
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6.1. Displacement and Area 219 draw
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6.1. Displacement and Area 221 It i
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6.1. Displacement and Area 223
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6.1. Displacement and Area 225 We d
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6.1. Displacement and Area 227 x i
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6.1. Displacement and Area 229 Both
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6.1. Displacement and Area 231 does
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6.2. The Fundamental Theorem of Cal
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6.3. Indefinite Integrals 243 Exerc
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6.3. Indefinite Integrals 245 Solut
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6.3. Indefinite Integrals 247 Exerc
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250 Techniques of Integration This
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252 Techniques of Integration ♣ E
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254 Techniques of Integration u, th
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256 Techniques of Integration 7.2 P
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258 Techniques of Integration = u7
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260 Techniques of Integration and
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262 Techniques of Integration ∫ N
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264 Techniques of Integration To in
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266 Techniques of Integration Exerc
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268 Techniques of Integration Expre
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270 Techniques of Integration The t
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272 Techniques of Integration Examp
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276 Techniques of Integration = sec
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278 Techniques of Integration Find
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280 Techniques of Integration Solut
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282 Techniques of Integration So al
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292 Techniques of Integration There
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294 Techniques of Integration Now u
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296 Techniques of Integration The f
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298 Techniques of Integration (e)
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302 Applications of Integration For
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304 Applications of Integration Sup
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306 Applications of Integration Gui
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308 Applications of Integration The
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310 Applications of Integration . y
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312 Applications of Integration . F
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314 Applications of Integration so
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316 Applications of Integration Exe
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318 Applications of Integration In
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320 Applications of Integration Sol
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322 Applications of Integration mag
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324 Applications of Integration . 1
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326 Applications of Integration and
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328 Applications of Integration Exe
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330 Applications of Integration Unf
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332 Applications of Integration Fig
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334 Applications of Integration is
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Chapter 9 Sequences and Series Cons
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9.1. Sequences 339 log 2 (1 + ε) >
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9.1. Sequences 341 Example 9.8: Con
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9.1. Sequences 343 below it is boun
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9.2. Series 345 If {kx n } ∞ n=0
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9.2. Series 347 for some number s.
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9.3. The Integral Test 349 If all o
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9.3. The Integral Test 351 Proof. W
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9.4. Alternating Series 353 Exercis
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9.5. Comparison Tests 355 Exercises
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9.5. Comparison Tests 357 Solution.
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9.7. The Ratio and Root Tests 359 E
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9.7. The Ratio and Root Tests 361 P
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9.8. Power Series 363 Definition 9.
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Exercise 9.8.2 Find the radius of c
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f ′′ (x)= f ′′′ (x)= ∞
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9.10. Taylor Series 369 Solution. T
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9.11. Taylor’s Theorem 371 a posi
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9.11. Taylor’s Theorem 373 Note t
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Chapter 10 Differential Equations M
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10.1. First Order Differential Equa
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10.1. First Order Differential Equa
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10.2. First Order Homogeneous Linea
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10.3. First Order Linear Equations
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10.4. Approximation 385 Exercises f
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10.4. Approximation 387 y .. 0.5 .
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10.5. Second Order Homogeneous Equa
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10.5. Second Order Homogeneous Equa
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10.6. Second Order Linear Equations
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Chapter 11 Polar Coordinates, Param
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11.1. Polar Coordinates 403 Solutio
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11.2. Slopes in Polar Coordinates 4
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11.3. Areas in Polar Coordinates 40
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11.3. Areas in Polar Coordinates 40
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11.4. Parametric Equations 411 Figu
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11.5 Calculus with Parametric Equat
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11.6. Conics in Polar Coordinates 4
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11.6. Conics in Polar Coordinates 4
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Chapter 12 Three Dimensions 12.1 Th
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12.1. The Coordinate System 421 Now
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12.2. Vectors 423 12.2 Vectors A ve
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12.2. Vectors 425 3.54 0.77 . . . .
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12.3. The Dot Product 427 Exercise
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12.3. The Dot Product 429 Solution.
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12.3. The Dot Product 431 v .. v ..
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12.4. The Cross Product 433 Exercis
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12.4. The Cross Product 435 We know
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12.5. Lines and Planes 437 Exercise
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12.5. Lines and Planes 439 Solution
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12.5. Lines and Planes 441 Example
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12.5. Lines and Planes 443 Exercise
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12.6. Other Coordinate Systems 445
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452 Partial Differentiation the lin
Page 470 and 471: 454 Partial Differentiation Exercis
Page 472 and 473: 456 Partial Differentiation Fortuna
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Page 482 and 483: 466 Partial Differentiation lim ε
Page 484 and 485: 468 Partial Differentiation 13.5 Di
Page 486 and 487: 470 Partial Differentiation Example
Page 492 and 493: 476 Partial Differentiation so we g
Page 494 and 495: 478 Partial Differentiation 0.0 0.2
Page 496 and 497: 480 Partial Differentiation and the
Page 498 and 499: 482 Partial Differentiation xz = 2y
Page 500 and 501: 484 Partial Differentiation x φ .
Page 502 and 503: 486 Multiple Integration point mult
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Page 506 and 507: 490 Multiple Integration 1 0 0 1 Fi
Page 508 and 509: 492 Multiple Integration Exercise 1
Page 512 and 513: 496 Multiple Integration This examp
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Page 516 and 517: 500 Multiple Integration Finally, M
Page 522 and 523: 506 Multiple Integration ∫ 1 = 1
Page 524 and 525: 508 Multiple Integration Solution.
Page 526 and 527: 510 Multiple Integration ∫ ∫
Page 528 and 529: .. 512 Multiple Integration As befo
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Page 532 and 533: 516 Multiple Integration ∫∫ Exe
Page 534 and 535: 518 Vector Functions separate funct
Page 536 and 537: 520 Vector Functions = 〈 f ′ (t
Page 538 and 539: 522 Vector Functions Figure 15.6:
Page 540 and 541: 524 Vector Functions 2.0 1.5 1.0 0.
Page 542 and 543: 526 Vector Functions Exercise 15.2.
Page 544 and 545: 528 Vector Functions (This integral
Page 546 and 547: 530 Vector Functions To remove the
Page 548 and 549: 532 Vector Functions Graphing this
Page 550 and 551: 534 Vector Functions Example 15.20
Page 552 and 553: 536 Vector Calculus which is the re
Page 554 and 555: 538 Vector Calculus Since f x = 2e
Page 556 and 557: 540 Vector Calculus Definition 16.4
Page 558 and 559: 542 Vector Calculus Example 16.8: W
Page 560 and 561: 544 Vector Calculus Proof. We write
Page 562 and 563: 546 Vector Calculus Exercise 16.3.2
Page 564 and 565: 548 Vector Calculus In this case, n
Page 566 and 567: 550 Vector Calculus We can now rewr
Page 568 and 569: 552 Vector Calculus 16.5 The Diverg
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554 Vector Calculus The remaining f
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556 Vector Calculus Suppose we inst
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558 Vector Calculus circle of radiu
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560 Vector Calculus Exercise 16.6.7
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562 Vector Calculus we might want t
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564 Vector Calculus where e is an e
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566 Vector Calculus ∫ b = ∫ = a
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Selected Exercise Answers 1.1.1 1.
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571 2.8.4 1. [2,3) ∪ (3,∞) 2. (
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573 4.5.18 −3(4 − x) 2 4.5.19 6
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575 5.1.15 500/ √ 3 − 200 ≈ 8
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577 5.6.21 min at x = 1 5.6.22 none
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579 7.1.2 x 5 /5 + 2x 3 /3 + x +C 7
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581 7.4.19 1/2e x2 +C 7.4.20 x 3 e
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583 7.7.20 diverges 7.7.21 diverges
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585 8.6.5 ¯x = 45/28, ȳ = 93/70 8
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587 9.8.2 R = e 10.1.2 y = arctant
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589 10.6.8 Ae t + Be 3t +(1/2)te 3t
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591 12.5.2 4(x + 1)+5(y − 2) −
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593 13.6.5 f x = 3cos(3x)cos(2y), f
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595 14.3.6 ¯x = 6/5, ȳ = 12/5 14.
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597 15.5.8 〈−3sint,2cost + 1/10
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Index absolute extrema, 172, 206 ab
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INDEX 601 at infinity, 87 indetermi
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Table of Integrals Table of Integra
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∫ 41. sin n ucos m udu= − sinn
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Integrals Involving u 2 − a 2 , a
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∫ du 108. u √ a + bu = √ 1
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