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SOLUTION 4 1 dx = 4 dx = 4 ln |x + 1| + C x+1 x+1 1 1 dx = ln | 2 + 3x| + C 2 + 3x 3 You Try It: Calculate the integral cos x dx 2 + sin x You Try It: Calculate the integral

The generic version of IComparable is declared like this: public interface IComparable<T> In this version, the type of data being compared is passed as a type argument to T This causes the declaration of CompareTo( ) to be changed, as shown next int CompareTo(T obj) Here, the type of data that CompareTo( ) operates on can be explicitly specified This makes IComparable<T> type-safe For this reason, IComparable<T> is now preferable to IComparable

IEquatable<T>is implemented by those classes that need to define how two objects should be compared for equality It defines only one method, Equals( ), which is shown here: bool Equals(T obj) The method returns true if obj is equal to the invoking object and false otherwise IEquatable<T> is implemented by several classes and structures in the NET Framework, including the numeric structures, Char, Int32, Boolean, and String

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111 INTRODUCTION 316 112DYNAMIC SYSTEM MODELING AND REDUCTION 317 1121 Natural Frequencies and Modes of Vibration 317 1122 Model Suf ciency and Model Reduction 321 113MASS, INERTIA, AND KINETIC ENERGY 324 1131 Finding Mass and Moment of Inertia 327 1132 Moving Moments of Inertia to Other Points 329 1133 Equivalent Mass or Inertia 329 114SPRINGS AND POTENTIAL ENERGY 332

The IConvertible interface is implemented by all of the value-type structures, string, and DateTime It specifies various type conversions Normally, classes that you create will not need to implement this interface

1 dx x [ln x]3/2

By implementing the ICloneable interface, you enable a copy of an object to be made ICloneable defines only one method, Clone( ), which is shown here: object Clone( ) This method makes a copy of the invoking object How you implement Clone( ) determines how the copy is made In general, there are two types of copies: deep and shallow When a deep copy is made, the copy and original are completely independent Thus, if the original object contained a reference to another object O, then a copy of O will also be made In a shallow copy, members are copied, but objects referred to by members are not If an object refers to some other object O, then after a shallow copy, both the copy and the original will refer to the same O, and any changes to O affect both the copy and the original Usually, you will implement Clone( ) so that it performs a deep copy Shallow copies can be made by using MemberwiseClone( ), which is defined by Object Here is an example that illustrates ICloneable It creates a class called Test that contains a reference to an object of a class called X Test uses Clone( ) to create a deep copy

1141 Coil Springs 334 1142 Compliance of Other Mechanical Elements 335 1143 Combinations of Springs 338 1144 Equivalent Springs and Mechanical Advantage 340 1145 Massive Springs 341 115DAMPERS AND DISSIPATION 342 1151 Viscous or Speed-Dependent Damping 343 1152 Coulomb or Dry Friction 344 1153 Mechanical Ef ciency 345 1154 Combinations and Equivalent Dampers 346 116EXAMPLE: MODELING AN AUTOMOTIVE VALVE-GEAR SYSTEM 348

// Demonstrate ICloneable using System; class X {

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Damping coef cient Rotary damping coef cient Mass moment of inertia about O (or moment of inertia of body O) K or k Stiffness or spring rate Kr Rotary stiffness or spring rate Free length of spring Lf M or m Mass N Number active coils vp Velocity vector at point P Magnitude of velocity at point P vp ro Vector from O to given point d De ection or relative displacement d Relative velocity Dq Angular de ection or displacement h Mechanical ef ciency w Angular velocity vector wn Natural frequency b br Jo lbf/(in/s) (N/(m/s)) ft-lbf/(rad/s) (Nm/(rad/s)) lbm-in2 (kgm2) lbf/in ft-lbf/rad in lbm in/s in/s in in in/s rad % rad/s rad/s or Hz (N/m) (Nm/rad) (m) (kg) (m/s) (m/s) (m) (m) (m/s) (rad) (rad/s) (rad/s or Hz)

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