Problems in Mathematical Analysis.pdf - pwp.net.ipl.pt

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174 Definite Integrals [Ch. 5 and Solution. We have: 10 p t* .-Jo.irt-o.i T = 250 metres =-=25 m/sec. 2. The work of a force. If a variable force X=f(x) acts in the direction of the x-axis, then the work of this force over an interval [x ly x z ] is A = Example 2. What work has to be performed to stretch a spring 6 cm, if a force of 1 kgf stretches it 1 by cm? Solution, According to Hook's law the force X kgf stretching the spring by xm is equal to X = kx, where k is a proportionality constant. Putting x = 0.01 m and X = l kgf, we get = 100 and, hence, X = 100v. Whence the sought-for work is 0.06 0.08 A = j 100 x dx = 50 x 2 = 0. 18 kgm 3. Kinetic energy. The kinetic energy of a material point of mass m and velocity v is defined as mv* The kinetic energy of a system of n material points with masses mv m 2% ..., m n having respective velocities t; lf v 2 , ..., v n , is equal to To compute the kinetic energy of a solid, the latter is appropriately partitioned into elementary particles (which play the part of material points); then by summing the kinetic energies of these particles we get, in the limit, an integral in place of the sum (1). Example 3. Find the kinetic energy of a homogeneous circular cylinder of density 6 with base radius R and altitude h rotating about its axis with angular velocity CD. Solution. For the elementary mass dm we take the mass of a hollow cylinder of altitude h with inner radius r and wall thickness dr (Fig. 60). We have: Since the linear velocity of the mass dm is equal to t; = /-co, the elementary kinetic energy is
Sec. 12] Applying Definite Integrals to Solution of Physical Problems 175 Whence r 9 dr^= nco 2 6/? 4 fc 4. Pressure of a liquid. To compute the force of liquid pressure we use Pascal's law, which states that the force of pressure of a liquid on an area S at a depth of immersion h Is where y is the specific weight of the liquid. Example 4. Find the force of pressure experienced by a semicircle of radius r submerged vertically in water so that its diameter is flush with the water surface (Fig 61). Solution, We partition the area of the semicircle into elements strips parallel to the surface of the water. The area of one such element (ignoring higher-order infinitesimals) located at a distance h from the surface is ds ^ 2xdh = 2 V'* h 2 dli. The pressure experienced by this element is where y is the specific weight of the water equal to unity. Whence the entire pressure is r = 2 C J 1751. The velocity of a body thrown vertically upwards with initial velocity V Q (air resistance neglected), is given by the
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TEXT FLY WITHIN THE BOOK ONLY
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OUP-43 30-1.71 5,000 OSMANIA UNIVER
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T. C. BapaneHKoe. C. M. Koean, r Jl
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TO THE READER MIR Publishers would
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Contents Sec. 6. Integrating Certai
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8 Chapter Contents Sec. 13. Linear
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Sec. 1. Functions Chapter I INTRODU
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Sec 1] Functions 13 8. Find the rat
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Sec. 1]_Functions_15 34. Prove that
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Sec. 2] Graphs of Elementary functi
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Sec. 2] Graphs of Elementary functi
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Sec. 2] Graphs of Elementary Functi
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Sec. 3] Limits 23 For the existence
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Sec. 3] 179. Hm (Vn + 1 \f~n). n -+
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Sec. #] Limits 27 209. lim 210. lim
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^Sec 3] Limits Therefore, Example 9
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Sec. 31 Limits 31 269. a) 270. lim-
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Sec. 4] Infinitely Small and Large
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Sec. 4] Infinitely Small and Large
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Sec. 5] Continuity of Functions 37
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Sec. 5]_Continuity of Functions_39
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Sec. 5] Continuity of Functions 334
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Sec. 1] Calculating Derivatives Dir
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Sec. 1] Calculating Derivatives Dir
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Sec. 2]__Tabular Differentiation_47
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Sec. 2]_Tabular Differentiation_49
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Sec. 2] Tabular Differentiation_ 51
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Sec. 2] Tabular Differentiation 504
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Sec. 2] Tabular Differentiation 55
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Sec. 3] The Derivatives of Function
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Sec. 3] The Derivatives of Function
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Sec 4] Geometrical and Mechanical A
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Sec. 4\ Geometrical and Mechanical
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Sec. 4] Geometrical and Mechanical
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Sec. 5]_Derivatives of Higher Order
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Sec. 5]_Derivatives of Higher Order
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Sec. 6] Differentials of First and
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Sec. 6}_Differentials of First and
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Sec. 7]_Mean-Value Theorems_75 745.
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Sec. 8] Taylor's Formula 77 765. a)
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Sec. 9]_ V Hospital-Bernoulli Rule
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Sec. 9] L' Hospital-Bernoulli Rule
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Chapter III THE EXTREMA OF A FUNCTI
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Sec. 1] The Extrema of a Function o
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Sec. 2]_The Direction of Concavity.
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Sec. 3]_Asymptotes_93 Example 2. Fi
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Sec. 3]_Asymptotes_95 Solution. Sin
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Sec 4] Graphing Functions by Charac
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CM CM -h CO O non ' > C " -~ a - >
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Sec. />] Differential of an Arc. Cu
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Sec 5] Differential of an Arc. Curv
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Sec. 5]_Differential of an Arc. Cur
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Sec. 1. Direct Integration 1. Basic
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Sec 1] Direct Integration_109 1048*
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Sec. 2] Integration by Substitution
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Sec. 2] Integration by Substitution
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Sec. 3\_Integration by Parts_H7 Exa
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Sec 4] Standard Integrals Containin
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Sec. 5] Integration of Rational Fun
Page 129 and 130: Sec. 5] In t egration of Rational F
Page 131 and 132: Sec. 6] Integrating Certain Irratio
Page 133 and 134: Sec. 6] Integrating Certain Irratio
Page 135 and 136: Sec. 7]_Integrating Trigonometric F
Page 137 and 138: Sec. 7]_Integrating Trigonometric F
Page 139 and 140: Sec. 8] Integration of Hyperbolic F
Page 141 and 142: Sec. 11] Using Reduction Formulas 1
Page 143 and 144: Sec. 12] Miscellaneous Examples on
Page 145 and 146: Sec. 1] The Definite Integral as th
Page 147 and 148: Sec. 2]_Evaluating Definite Integra
Page 149 and 150: Sec. 3]_Improper Integrals_143 Sec.
Page 151 and 152: Sec. 3] Improper Integrals 145 Solu
Page 153 and 154: Sec. 4] Change of Variable in a Def
Page 155 and 156: Sec. 5] Integration by Parts 149 Bu
Page 157 and 158: Sec. 6] Mean-Value Theorem 151 In p
Page 159 and 160: Sec. 7] The Areas of Plane Figures
Page 161 and 162: Sec. 7] The Areas of Plane Figures
Page 163 and 164: Sec. 7\_The Areas of Plane Figures_
Page 165 and 166: Sec. 8] The Arc Length of a Curve 1
Page 167 and 168: Sec, 9] Volumes of Solids 161 1673.
Page 169 and 170: Sec. 9] Volumes of Solids 163' Solu
Page 171 and 172: Sec. 9]_Volumes of Solids_165 1693.
Page 173 and 174: Sec. 10] The Area of a Surface of R
Page 175 and 176: Sec. II] Moments. Centres of Gravit
Page 177 and 178: Sec 11] Moments. Centres of Gravity
Page 179: &?c. 12] Applying Definite Integral
Page 183 and 184: Sec. 12] Applying Definite Integral
Page 185 and 186: Sec. 12] Applying Definite Integral
Page 187 and 188: Sec. 1) Basic Notions [SI Putting *
Page 189 and 190: Sec. 1]_Basic Notions_183 1790*. Le
Page 191 and 192: Sec. 3] Partial Derivatives 185 180
Page 193 and 194: Sec. 4]_Total Differential of a Fun
Page 195 and 196: Sec. 4}_Total Differential of a Fun
Page 197 and 198: Sec. 5]_Differentiation of Composit
Page 199 and 200: Sec. 6]_Derivative in a Given Direc
Page 201 and 202: Sec. 6] Derivative in a Given Direc
Page 203 and 204: Sec. 7]_Higher-Order Derivatives an
Page 205 and 206: Sec. 7] Higher-Order Derivatives an
Page 207 and 208: Sec. 7]_Higher-Order Derivatives an
Page 209 and 210: Sec. ti\_Integration of Total Diffe
Page 211 and 212: Sec. 9]_Differentiation of Implicit
Page 213 and 214: Sec. 9] Differentiation of Implicit
Page 215 and 216: Sec. 9]_Differentiation of Implicit
Page 217 and 218: Sec. 10]_Change of Variables_2H 196
Page 219 and 220: Sec. 10]_Change of Variables_213 or
Page 221 and 222: Sec. 10] Change of Variables 215 an
Page 223 and 224: Sec. 11]_The Tangent Plane and the
Page 225 and 226: Sec. 11]_The Tangent Plane and the
Page 227 and 228: Sec. 12] Taylor's Formula for a Fun
Page 229 and 230: Sec. 13]_The Extremum of a Function
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Sec. 13] The Extremutn of a Functio
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Sec. 14] Finding the Greatest and S
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Sec. 14] Finding the Greatest and S
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Sec. 15] Singular Points of Plane C
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Sec. 16] Envelope 233 Solution. The
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Sec. 18] The Vector Function of a S
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Sec. 18]_The Vector Function of a S
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Sec. 19] The Natural Trihedron of a
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Sec. 19] The Natural Trihedron of a
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Sec. 20] Curvature and Torsion of a
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Sec. 20]_Curvature and Torsion of a
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Sec. 1] The Double Integral in Rect
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Sec. 2]_Change of Variables in a Do
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Sec. 2]_Change of Variables in a Do
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Sec 3]_Computing Areas_257 2177. Co
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Sec. 5]_Computing th Area* of Surfa
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Sec. 6]_Applications of the Double
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Sec. 7] Triple Integrals 2o3 Evalua
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Sec 7\ Triple Integrals 265 We ther
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gee. 7]_Triple Integrals_267 2251.
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Sec. 8] Improper Integrals Dependen
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Sec. 8]_Improper Integrals Dependen
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See. 9] Line Integrals 273 Test for
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Sec. 9]____Line Inlegtah____275 whe
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Sec. 9]_Line Integrals__277 express
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Sec. 9] Lim Integrals 279 (/-axis;
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Sec 9] Line Integrals 281 Evaluate
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Sec. 9] Line Integrals 2S& D. Appli
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Sec. 10] Surface Integrals 285 When
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ec. 11]_The Ostrogradsky-Gauss Form
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Sec. 12] Fundamentals of Field Theo
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Sec. 12] Fundamentals of Field Theo
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Sec. 1. Number Series Chapter VIII
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Sec. 1]_Number Series_295 c) D'Alem
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Sec /\_Number Series_297 Note. For
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2418. + + 0419 J___1_ . 4- 4- 4- '
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Sec. 1] Number Series 2464. cos-2-)
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Sec. 1]_Number Series_303 - V r(2-Q
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Sec. 2]_Functional Series_305 In th
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Sec. 2]_Functional Series_307 [the
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Sec. 2] Functional Series 309 Deter
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Sec. 3]_Taylor's Series_311 v8 s r
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Sec. 3] Taylor's Series 313 Since a
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Sec 3} Taylor's Series 315 2615. ln
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Sec. 3] Taylor's Series 317 2651. F
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Sec. 4] Fourier Series 319 If a fun
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Sec. 4\__Fourier Series _321 2690.
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Sec 1 1 Verifying Solutions 323 It
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Sec. 21 First-Order Differential Eq
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Sec. 3] Differential Equations with
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Sec. 3\ Differential Equations with
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Sec. 4]_First-Order Homogeneous Dif
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Sec. 5] Bernoulli's Equation 333 Co
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Sec. 6] Exact Differential Equation
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Sec. 7] First-Order Differential Eq
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Sec 8]_The Lagrange and Clairaut Eq
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Sec. 9] Miscellaneous Exercises on
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Sec. 9] Miscellaneous Exercises on
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Sec. 10] Higher-Order Differential
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Sec. 10]__Higher-Order Differential
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Sec. 11]_Linear Differential Equati
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Sec. 12] Linear Differential Equati
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Sec. 14]_Euler's Equations__357 Fin
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Sec. 15]_Systems of Differential Eq
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.Sec. 16} Integration of Differenti
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Sec. 17] Problems on Fourier's Meth
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Sec. 17]_Problems on Fourier's Meth
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Chapter X APPROXIMATE CALCULATIONS
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Sec. 1] Operations on Approximate N
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Sec. /]_Operations on Approximate N
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Sec, 2] Interpolation of Functions
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Sec. 2] Interpolation of Functions
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Sec. 3]_Computing the Real Roots of
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Sec. 4] Numerical Integration of Fu
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Sec. 5] Numerical Integration of Or
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Table 3. Basic Table for Calculatin
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394 Approximate Calculations \Ch 10
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ANSWERS Chapter I 1. Solution. Sinc
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398 Answers Form a table of values:
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400 Answer* 271. Solution. If x & k
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402 Answers b) /'(!)= lim 1^15^= li
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404_Answers 494. , . 495. - - . 1 2
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-406 Answers 640. Hint. The equatio
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408_Answers_ 755. e xcosa sin(*sina
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410_Answers_4 where R is the radius
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412 Answers 1/~JT f Q \ J/max = TT=
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414 Answers 976. j/mi n =1.32 when
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416 Answers 1037. j/ 1040. 3x* 13 J
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418 1 1137. 4- 6 3a . In I ocot3x|
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420 Answers 4^8 Problem 1218*. 1221
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422 Answers X arctan(* + 2). 1294.
Page 430 and 431:
424 Answers sin 5* , 1374. -!- 1377
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426 Answers 3 1442. In l-x2 ' '" 2
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428_Answers_ the function /(x) = --
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430 Answers 1651. 3na 2 . 1652. n(b
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432 Answers inertia /=Y 2jttf 1 r *
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434 Answers along the large lower s
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436 Answers cos^. ,810. * ox x* x d
Page 444 and 445:
438_Answers 1905. 1914. u(*. !/) =
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440 Answers 1983. 3* + 4j/-H2z 169
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442 Answers is ~ |/2 + -~=, the alt
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444 Answers (binormal). 2103. bx *=
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444_Answers_x = \ ' > (principal no
Page 454 and 455:
446 Answers 2 a a a 2139. ( dy(f(x,
Page 456 and 457:
448_Answers_ r 4 **/* [ rj-cos*(p--
Page 458 and 459:
450 Answers 2243. 2244. 1 i VT=T* ^
Page 460 and 461:
452 Answers 2 t 2297. ) ~ll . |-[(I
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454_Answers__ TtD the flux, use the
Page 464 and 465:
456__Answers_ and when x | ^ =- , t
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458 Answers
Page 468 and 469:
460 Answers %-l n=o (n)=^=%. 2873.
Page 470 and 471:
462 Answers Chapter IX 2704. Yes. 2
Page 472 and 473:
464 Answers 2822. g-C*+; y=2*. 2824
Page 474 and 475:
466 Answers x + C z . 2914. #=
Page 476 and 477:
468_Answers_ A $007. 1) x = C cosco
Page 478 and 479:
470 Answers 3081. x = C l e 3082. x
Page 480 and 481:
472 Answers ,*. f (2"+ 1)JtJf dx. H
Page 482 and 483:
474 Answers 3131. a) 0.211; 0.389;
Page 484 and 485:
476 Appendix III. Inverse Quantitie
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478 Appendix IV. Trigonometric Func
Page 488 and 489:
480 Appendix VI. Some Curves (for R
Page 490 and 491:
482 Appendix 11. Graphs //-=secx an
Page 492 and 493:
486 Appendix 24. Strophoid, * u s^-
Page 494 and 495:
Absolute error Absolute value 367 o
Page 496 and 497:
490 Index Differential equations of
Page 498 and 499:
492 Index Infinite discontinuities
Page 500 and 501:
494 Index Operator Hamiltonian 288
Page 502:
496 Index Taylor's series Term 311,
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