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Unformatted text preview: ''JUST THE MATHS'' by A.J. Hobson TEACHING UNITS - TABLE OF CONTENTS (Average number of pages = 1038 ¸140 = 7.4 per unit) All units are in presented as .PDF files [Home] [Foreword] [About the Author] UNIT 1.1 - ALGEBRA 1 - INTRODUCTION TO ALGEBRA 1.1.1 The Language of Algebra 1.1.2 The Laws of Algebra 1.1.3 Priorities in Calculations 1.1.4 Factors 1.1.5 Exercises 1.1.6 Answers to exercises (6 pages) UNIT 1.2 - ALGEBRA 2 - NUMBERWORK 1.2.1 Types of number 1.2.2 Decimal numbers 1.2.3 Use of electronic calculators 1.2.4 Scientific notation 1.2.5 Percentages 1.2.6 Ratio 1.2.7 Exercises 1.2.8 Answers to exercises (8 pages) UNIT 1.3 - ALGEBRA 3 - INDICES AND RADICALS (OR SURDS) 1.3.1 Indices 1.3.2 Radicals (or Surds) 1.3.3 Exercises 1.3.4 Answers to exercises (8 pages) UNIT 1.4 - ALGEBRA 4 - LOGARITHMS 1.4.1 Common logarithms 1.4.2 Logarithms in general 1.4.3 Useful Results 1.4.4 Properties of logarithms 1.4.5 Natural logarithms 1.4.6 Graphs of logarithmic and exponential functions 1.4.7 Logarithmic scales 1.4.8 Exercises 1.4.9 Answers to exercises (10 pages) UNIT 1.5 - ALGEBRA 5 - MANIPULATION OF ALGEBRAIC EXPRESSIONS 1.5.1 Simplification of expressions 1.5.2 Factorisation 1 of 20 1.5.3 Completing the square in a quadratic expression 1.5.4 Algebraic Fractions 1.5.5 Exercises 1.5.6 Answers to exercises (9 pages) UNIT 1.6 - ALGEBRA 6 - FORMULAE AND ALGEBRAIC EQUATIONS 1.6.1 Transposition of formulae 1.6.2 Solution of linear equations 1.6.3 Solution of quadratic equations 1.6.4 Exercises 1.6.5 Answers to exercises (7 pages) UNIT 1.7 - ALGEBRA 7 - SIMULTANEOUS LINEAR EQUATIONS 1.7.1 Two simultaneous linear equations in two unknowns 1.7.2 Three simultaneous linear equations in three unknowns 1.7.3 Ill-conditioned equations 1.7.4 Exercises 1.7.5 Answers to exercises (6 pages) UNIT 1.8 - ALGEBRA 8 - POLYNOMIALS 1.8.1 The factor theorem 1.8.2 Application to quadratic and cubic expressions 1.8.3 Cubic equations 1.8.4 Long division of polynomials 1.8.5 Exercises 1.8.6 Answers to exercises (8 pages) UNIT 1.9 - ALGEBRA 9 - THE THEORY OF PARTIAL FRACTIONS 1.9.1 Introduction 1.9.2 Standard types of partial fraction problem 1.9.3 Exercises 1.9.4 Answers to exercises (7 pages) UNIT 1.10 - ALGEBRA 10 - INEQUALITIES 1 1.10.1 Introduction 1.10.2 Algebraic rules for inequalities 1.10.3 Intervals 1.10.4 Exercises 1.10.5 Answers to exercises (5 pages) UNIT 1.11 - ALGEBRA 11 - INEQUALITIES 2 1.11.1 Recap on modulus, absolute value or numerical value 1.11.2 Interval inequalities 1.11.3 Exercises 1.11.4 Answers to exercises (5 pages) UNIT 2.1 - SERIES 1 - ELEMENTARY PROGRESSIONS AND SERIES 2.1.1 Arithmetic progressions 2.1.2 Arithmetic series 2.1.3 Geometric progressions 2.1.4 Geometric series 2.1.5 More general progressions and series 2.1.6 Exercises 2 of 20 2.1.7 Answers to exercises (12 pages) UNIT 2.2 - SERIES 2 - BINOMIAL SERIES 2.2.1 Pascal's Triangle 2.2.2 Binomial Formulae 2.2.3 Exercises 2.2.4 Answers to exercises (9 pages) UNIT 2.3 - SERIES 3 - ELEMENTARY CONVERGENCE AND DIVERGENCE 2.3.1 The definitions of convergence and divergence 2.3.2 Tests for convergence and divergence (positive terms) 2.3.3 Exercises 2.3.4 Answers to exercises (13 pages) UNIT 2.4 - SERIES 4 - FURTHER CONVERGENCE AND DIVERGENCE 2.4.1 Series of positive and negative terms 2.4.2 Absolute and conditional convergence 2.4.3 Tests for absolute convergence 2.4.4 Power series 2.4.5 Exercises 2.4.6 Answers to exercises (9 pages) UNIT 3.1 - TRIGONOMETRY 1 - ANGLES AND TRIGONOMETRIC FUNCTIONS 3.1.1 Introduction 3.1.2 Angular measure 3.1.3 Trigonometric functions 3.1.4 Exercises 3.1.5 Answers to exercises (6 pages) UNIT 3.2 - TRIGONOMETRY 2 - GRAPHS OF TRIGONOMETRIC FUNCTIONS 3.2.1 Graphs of elementary trigonometric functions 3.2.2 Graphs of more general trigonometric functions 3.2.3 Exercises 3.2.4 Answers to exercises (7 pages) UNIT 3.3 - TRIGONOMETRY 3 - APPROXIMATIONS AND INVERSE FUNCTIONS 3.3.1 Approximations for trigonometric functions 3.3.2 Inverse trigonometric functions 3.3.3 Exercises 3.3.4 Answers to exercises (6 pages) UNIT 3.4 - TRIGONOMETRY 4 - SOLUTION OF TRIANGLES 3.4.1 Introduction 3.4.2 Right-angled triangles 3.4.3 The sine and cosine rules 3.4.4 Exercises 3.4.5 Answers to exercises (5 pages) UNIT 3.5 - TRIGONOMETRY 5 - TRIGONOMETRIC IDENTITIES AND WAVE-FORMS 3.5.1 Trigonometric identities 3.5.2 Amplitude, wave-length, frequency and phase-angle 3.5.3 Exercises 3 of 20 3.5.4 Answers to exercises (8 pages) UNIT 4.1 - HYPERBOLIC FUNCTIONS 1 - DEFINITIONS, GRAPHS AND IDENTITIES 4.1.1 Introduction 4.1.2 Definitions 4.1.3 Graphs of hyperbolic functions 4.1.4 Hyperbolic identities 4.1.5 Osborn's rule 4.1.6 Exercises 4.1.7 Answers to exercises (7 pages) UNIT 4.2 - HYPERBOLIC FUNCTIONS 2 - INVERSE HYPERBOLIC FUNCTIONS 4.2.1 Introduction 4.2.2 The proofs of the standard formulae 4.2.3 Exercises 4.2.4 Answers to exercises (6 pages) UNIT 5.1 - GEOMETRY 1 - CO-ORDINATES, DISTANCE AND GRADIENT 5.1.1 Co-ordinates 5.1.2 Relationship between polar & cartesian co-ordinates 5.1.3 The distance between two points 5.1.4 Gradient 5.1.5 Exercises 5.1.6 Answers to exercises (5 pages) UNIT 5.2 - GEOMETRY 2 - THE STRAIGHT LINE 5.2.1 Preamble 5.2.2 Standard equations of a straight line 5.2.3 Perpendicular straight lines 5.2.4 Change of origin 5.2.5 Exercises 5.2.6 Answers to exercises (8 pages) UNIT 5.3 - GEOMETRY 3 - STRAIGHT LINE LAWS 5.3.1 Introduction 5.3.2 Laws reducible to linear form 5.3.3 The use of logarithmic graph paper 5.3.4 Exercises 5.3.5 Answers to exercises (7 pages) UNIT 5.4 - GEOMETRY 4 - ELEMENTARY LINEAR PROGRAMMING 5.4.1 Feasible Regions 5.4.2 Objective functions 5.4.3 Exercises 5.4.4 Answers to exercises (9 pages) UNIT 5.5 - GEOMETRY 5 - CONIC SECTIONS (THE CIRCLE) 5.5.1 Introduction 5.5.2 Standard equations for a circle 5.5.3 Exercises 5.5.4 Answers to exercises (5 pages) UNIT 5.6 - GEOMETRY 6 - CONIC SECTIONS (THE PARABOLA) 4 of 20 5.6.1 Introduction (the standard parabola) 5.6.2 Other forms of the equation of a parabola 5.6.3 Exercises 5.6.4 Answers to exercises (6 pages) UNIT 5.7 - GEOMETRY 7 - CONIC SECTIONS (THE ELLIPSE) 5.7.1 Introduction (the standard ellipse) 5.7.2 A more general form for the equation of an ellipse 5.7.2 Exercises 5.7.3 Answers to exercises (4 pages) UNIT 5.8 - GEOMETRY 8 - CONIC SECTIONS (THE HYPERBOLA) 5.8.1 Introduction (the standard hyperbola) 5.8.2 Asymptotes 5.8.3 More general forms for the equation of a hyperbola 5.8.4 The rectangular hyperbola 5.8.5 Exercises 5.8.6 Answers to exercises (8 pages) UNIT 5.9 - GEOMETRY 9 - CURVE SKETCHING IN GENERAL 5.9.1 Symmetry 5.9.2 Intersections with the co-ordinate axes 5.9.3 Restrictions on the range of either variable 5.9.4 The form of the curve near the origin 5.9.5 Asymptotes 5.9.6 Exercises 5.9.7 Answers to exercises (10 pages) UNIT 5.10 - GEOMETRY 10 - GRAPHICAL SOLUTIONS 5.10.1 The graphical solution of linear equations 5.10.2 The graphical solution of quadratic equations 5.10.3 The graphical solution of simultaneous equations 5.10.4 Exercises 5.10.5 Answers to exercises (7 pages) UNIT 5.11 - GEOMETRY 11 - POLAR CURVES 5.11.1 Introduction 5.11.2 The use of polar graph paper 5.11.3 Exercises 5.11.4 Answers to exercises (10 pages) UNIT 6.1 - COMPLEX NUMBERS 1 - DEFINITIONS AND ALGEBRA 6.1.1 The definition of a complex number 6.1.2 The algebra of complex numbers 6.1.3 Exercises 6.1.4 Answers to exercises (8 pages) UNIT 6.2 - COMPLEX NUMBERS 2 - THE ARGAND DIAGRAM 6.2.1 Introduction 6.2.2 Graphical addition and subtraction 6.2.3 Multiplication by j 6.2.4 Modulus and argument 6.2.5 Exercises 5 of 20 6.2.6 Answers to exercises (7 pages) UNIT 6.3 - COMPLEX NUMBERS 3 - THE POLAR AND EXPONENTIAL FORMS 6.3.1 The polar form 6.3.2 The exponential form 6.3.3 Products and quotients in polar form 6.3.4 Exercises 6.3.5 Answers to exercises (8 pages) UNIT 6.4 - COMPLEX NUMBERS 4 - POWERS OF COMPLEX NUMBERS 6.4.1 Positive whole number powers 6.4.2 Negative whole number powers 6.4.3 Fractional powers & De Moivre's Theorem 6.4.4 Exercises 6.4.5 Answers to exercises (5 pages) UNIT 6.5 - COMPLEX NUMBERS 5 - APPLICATIONS TO TRIGONOMETRIC IDENTITIES 6.5.1 Introduction 6.5.2 Expressions for cosn q, sinn q in terms of cosq, sinq n n 6.5.3 Expressions for cos q and sin q in terms of sines and cosines of whole multiples of x 6.5.4 Exercises 6.5.5 Answers to exercises (5 pages) UNIT 6.6 - COMPLEX NUMBERS 6 - COMPLEX LOCI 6.6.1 Introduction 6.6.2 The circle 6.6.3 The half-straight-line 6.6.4 More general loci 6.6.5 Exercises 6.6.6 Answers to exercises (6 pages) UNIT 7.1 - DETERMINANTS 1 - SECOND ORDER DETERMINANTS 7.1.1 Pairs of simultaneous linear equations 7.1.2 The definition of a second order determinant 7.1.3 Cramer's Rule for two simultaneous linear equations 7.1.4 Exercises 7.1.5 Answers to exercises (7 pages) UNIT 7.2 - DETERMINANTS 2 - CONSISTENCY AND THIRD ORDER DETERMINANTS 7.2.1 Consistency for three simultaneous linear equations in two unknowns 7.2.2 The definition of a third order determinant 7.2.3 The rule of Sarrus 7.2.4 Cramer's rule for three simultaneous linear equations in three unknowns 7.2.5 Exercises 7.2.6 Answers to exercises (10 pages) UNIT 7.3 - DETERMINANTS 3 - FURTHER EVALUATION OF 3 X 3 DETERMINANTS 7.3.1 Expansion by any row or column 7.3.2 Row and column operations on determinants 7.3.3 Exercises 7.3.4 Answers to exercises (10 pages) 6 of 20 UNIT 7.4 - DETERMINANTS 4 - HOMOGENEOUS LINEAR EQUATIONS 7.4.1 Trivial and non-trivial solutions 7.4.2 Exercises 7.4.3 Answers to exercises (7 pages) UNIT 8.1 - VECTORS 1 - INTRODUCTION TO VECTOR ALGEBRA 8.1.1 Definitions 8.1.2 Addition and subtraction of vectors 8.1.3 Multiplication of a vector by a scalar 8.1.4 Laws of algebra obeyed by vectors 8.1.5 Vector proofs of geometrical results 8.1.6 Exercises 8.1.7 Answers to exercises (7 pages) UNIT 8.2 - VECTORS 2 - VECTORS IN COMPONENT FORM 8.2.1 The components of a vector 8.2.2 The magnitude of a vector in component form 8.2.3 The sum and difference of vectors in component form 8.2.4 The direction cosines of a vector 8.2.5 Exercises 8.2.6 Answers to exercises (6 pages) UNIT 8.3 - VECTORS 3 - MULTIPLICATION OF ONE VECTOR BY ANOTHER 8.3.1 The scalar product (or 'dot' product) 8.3.2 Deductions from the definition of dot product 8.3.3 The standard formula for dot product 8.3.4 The vector product (or 'cross' product) 8.3.5 Deductions from the definition of cross product 8.3.6 The standard formula for cross product 8.3.7 Exercises 8.3.8 Answers to exercises (8 pages) UNIT 8.4 - VECTORS 4 - TRIPLE PRODUCTS 8.4.1 The triple scalar product 8.4.2 The triple vector product 8.4.3 Exercises 8.4.4 Answers to exercises (7 pages) UNIT 8.5 - VECTORS 5 - VECTOR EQUATIONS OF STRAIGHT LINES 8.5.1 Introduction 8.5.2 The straight line passing through a given point and parallel to a given vector 8.5.3 The straight line passing through two given points 8.5.4 The perpendicular distance of a point from a straight line 8.5.5 The shortest distance between two parallel straight lines 8.5.6 The shortest distance between two skew straight lines 8.5.7 Exercises 8.5.8 Answers to exercises (14 pages) UNIT 8.6 - VECTORS 6 - VECTOR EQUATIONS OF PLANES 8.6.1 The plane passing through a given point and perpendicular to a given vector 8.6.2 The plane passing through three given points 8.6.3 The point of intersection of a straight line and a plane 8.6.4 The line of intersection of two planes 7 of 20 8.6.5 The perpendicular distance of a point from a plane 8.6.6 Exercises 8.6.7 Answers to exercises (9 pages) UNIT 9.1 - MATRICES 1 - DEFINITIONS AND ELEMENTARY MATRIX ALGEBRA 9.1.1 Introduction 9.1.2 Definitions 9.1.3 The algebra of matrices (part one) 9.1.4 Exercises 9.1.5 Answers to exercises (8 pages) UNIT 9.2 - MATRICES 2 - FURTHER MATRIX ALGEBRA 9.2.1 Multiplication by a single number 9.2.2 The product of two matrices 9.2.3 The non-commutativity of matrix products 9.2.4 Multiplicative identity matrices 9.2.5 Exercises 9.2.6 Answers to exercises (6 pages) UNIT 9.3 - MATRICES 3 - MATRIX INVERSION AND SIMULTANEOUS EQUATIONS 9.3.1 Introduction 9.3.2 Matrix representation of simultaneous linear equations 9.3.3 The definition of a multiplicative inverse 9.3.4 The formula for a multiplicative inverse 9.3.5 Exercises 9.3.6 Answers to exercises (11 pages) UNIT 9.4 - MATRICES 4 - ROW OPERATIONS 9.4.1 Matrix inverses by row operations 9.4.2 Gaussian elimination (the elementary version) 9.4.3 Exercises 9.4.4 Answers to exercises (10 pages) UNIT 9.5 - MATRICES 5 - CONSISTENCY AND RANK 9.5.1 The consistency of simultaneous linear equations 9.5.2 The row-echelon form of a matrix 9.5.3 The rank of a matrix 9.5.4 Exercises 9.5.5 Answers to exercises (9 pages) UNIT 9.6 - MATRICES 6 - EIGENVALUES AND EIGENVECTORS 9.6.1 The statement of the problem 9.6.2 The solution of the problem 9.6.3 Exercises 9.6.4 Answers to exercises (9 pages) UNIT 9.7 - MATRICES 7 - LINEARLY INDEPENDENT AND NORMALISED EIGENVECTORS 9.7.1 Linearly independent eigenvectors 9.7.2 Normalised eigenvectors 9.7.3 Exercises 9.7.4 Answers to exercises (5 pages) UNIT 9.8 - MATRICES 8 - CHARACTERISTIC PROPERTIES AND SIMILARITY 8 of 20 TRANSFORMATIONS 9.8.1 Properties of eigenvalues and eigenvectors 9.8.2 Similar matrices 9.8.3 Exercises 9.7.4 Answers to exercises (9 pages) UNIT 9.9 - MATRICES 9 - MODAL AND SPECTRAL MATRICES 9.9.1 Assumptions and definitions 9.9.2 Diagonalisation of a matrix 9.9.3 Exercises 9.9.4 Answers to exercises (9 pages) UNIT 9.10 - MATRICES 10 - SYMMETRIC MATRICES AND QUADRATIC FORMS 9.10.1 Symmetric matrices 9.10.2 Quadratic forms 9.10.3 Exercises 9.10.4 Answers to exercises (7 pages) UNIT 10.1 - DIFFERENTIATION 1 - FUNTIONS AND LIMITS 10.1.1 Functional notation 10.1.2 Numerical evaluation of functions 10.1.3 Functions of a linear function 10.1.4 Composite functions 10.1.5 Indeterminate forms 10.1.6 Even and odd functions 10.1.7 Exercises 10.1.8 Answers to exercises (12 pages) UNIT 10.2 - DIFFERENTIATION 2 - RATES OF CHANGE 10.2.1 Introduction 10.2.2 Average rates of change 10.2.3 Instantaneous rates of change 10.2.4 Derivatives 10.2.5 Exercises 10.2.6 Answers to exercises (7 pages) UNIT 10.3 - DIFFERENTIATION 3 - ELEMENTARY TECHNIQUES OF DIFFERENTIATION 10.3.1 Standard derivatives 10.3.2 Rules of differentiation 10.3.3 Exercises 10.3.4 Answers to exercises (9 pages) UNIT 10.4 - DIFFERENTIATION 4 - PRODUCTS, QUOTIENTS AND LOGARITHMIC DIFFERENTIATION 10.4.1 Products 10.4.2 Quotients 10.4.3 Logarithmic differentiation 10.4.4 Exercises 10.4.5 Answers to exercises (10 pages) UNIT 10.5 - DIFFERENTIATION 5 - IMPLICIT AND PARAMETRIC FUNCTIONS 10.5.1 Implicit functions 10.5.2 Parametric functions 9 of 20 10.5.3 Exercises 10.5.4 Answers to exercises (5 pages) UNIT 10.6 - DIFFERENTIATION 6 - DERIVATIVES OF INVERSE TRIGONOMETRIC FUNCTIONS 10.6.1 Summary of results 10.6.2 The derivative of an inverse sine 10.6.3 The derivative of an inverse cosine 10.6.4 The derivative of an inverse tangent 10.6.5 Exercises 10.6.6 Answers to exercises (7 pages) UNIT 10.7 - DIFFERENTIATION 7 - DERIVATIVES OF INVERSE HYPERBOLIC FUNCTIONS 10.7.1 Summary of results 10.7.2 The derivative of an inverse hyperbolic sine 10.7.3 The derivative of an inverse hyperbolic cosine 10.7.4 The derivative of an inverse hyperbolic tangent 10.7.5 Exercises 10.7.6 Answers to exercises (7 pages) UNIT 10.8 - DIFFERENTIATION 8 - HIGHER DERVIVATIVES 10.8.1 The theory 10.8.2 Exercises 10.8.3 Answers to exercises (4 pages) UNIT 11.1 - DIFFERENTIATION APPLICATIONS 1 - TANGENTS AND NORMALS 11.1.1 Tangents 11.1.2 Normals 11.1.3 Exercises 11.1.4 Answers to exercises (5 pages) UNIT 11.2 - DIFFERENTIATION APPLICATIONS 2 - LOCAL MAXIMA, LOCAL MINIMA AND POINTS OF INFLEXION 11.2.1 Introduction 11.2.2 Local maxima 11.2.3 Local minima 11.2.4 Points of inflexion 11.2.5 The location of stationary points and their nature 11.2.6 Exercises 11.2.7 Answers to exercises (14 pages) UNIT 11.3 - DIFFERENTIATION APPLICATIONS 3 - CURVATURE 11.3.1 Introduction 11.3.2 Curvature in cartesian co-ordinates 11.3.3 Exercises 11.3.4 Answers to exercises (6 pages) UNIT 11.4 - DIFFERENTIATION APPLICATIONS 4 - CIRCLE, RADIUS AND CENTRE OF CURVATURE 11.4.1 Introduction 11.4.2 Radius of curvature 11.4.3 Centre of curvature 11.4.4 Exercises 11.4.5 Answers to exercises (5 pages) 10 of 20 UNIT 11.5 - DIFFERENTIATION APPLICATIONS 5 - MACLAURIN'S AND TAYLOR'S SERIES 11.5.1 Maclaurin's series 11.5.2 Standard series 11.5.3 Taylor's series 11.5.4 Exercises 11.5.5 Answers to exercises (10 pages) UNIT 11.6 - DIFFERENTIATION APPLICATIONS 6 - SMALL INCREMENTS AND SMALL ERRORS 11.6.1 Small increments 11.6.2 Small errors 11.6.3 Exercises 11.6.4 Answers to exercises (8 pages) UNIT 12.1 - INTEGRATION 1 - ELEMENTARY INDEFINITE INTEGRALS 12.1.1 The definition of an integral 12.1.2 Elementary techniques of integration 12.1.3 Exercises 12.1.4 Answers to exercises (11 pages) UNIT 12.2 - INTEGRATION 2 - INTRODUCTION TO DEFINITE INTEGRALS 12.2.1 Definition and examples 12.2.2 Exercises 12.2.3 Answers to exercises (3 pages) UNIT 12.3 - INTEGRATION 3 - THE METHOD OF COMPLETING THE SQUARE 12.3.1 Introduction and examples 12.3.2 Exercises 12.3.3 Answers to exercises (4 pages) UNIT 12.4 - INTEGRATION 4 - INTEGRATION BY SUBSTITUTION IN GENERAL 12.4.1 Examples using the standard formula 12.4.2 Integrals involving a function and its derivative 12.4.3 Exercises 12.4.4 Answers to exercises (5 pages) UNIT 12.5 - INTEGRATION 5 - INTEGRATION BY PARTS 12.5.1 The standard formula 12.5.2 Exercises 12.5.3 Answers to exercises (6 pages) UNIT 12.6 - INTEGRATION 6 - INTEGRATION BY PARTIAL FRACTIONS 12.6.1 Introduction and illustrations 12.6.2 Exercises 12.6.3 Answers to exercises (4 pages) UNIT 12.7 - INTEGRATION 7 - FURTHER TRIGONOMETRIC FUNCTIONS 12.7.1 Products of sines and cosines 12.7.2 Powers of sines and cosines 12.7.3 Exercises 12.7.4 Answers to exercises (7 pages) UNIT 12.8 - INTEGRATION 8 - THE TANGENT SUBSTITUTIONS 12.8.1 The substitution t = tanx 12.8.2 The substitution t = tan(x/2) 11 of 20 12.8.3 Exercises 12.8.4 Answers to exercises (5 pages) UNIT 12.9 - INTEGRATION 9 - REDUCTION FORMULAE 12.9.1 Indefinite integrals 12.9.2 Definite integrals 12.9.3 Exercises 12.9.4 Answers to exercises (7 pages) UNIT 12.10 - INTEGRATION 10 - FURTHER REDUCTION FORMULAE 12.10.1 Integer powers of a sine 12.10.2 Integer powers of a cosine 12.10.3 Wallis's formulae 12.10.4 Combinations of sines and cosines 12.10.5 Exercises 12.10.6 Answers to exercises (8 pages) UNIT 13.1 - INTEGRATION APPLICATIONS 1 - THE AREA UNDER A CURVE 13.1.1 The elementary formula 13.1.2 Definite integration as a summation 13.1.3 Exercises 13.1.4 Answers to exercises (6 pages) UNIT 13.2 - INTEGRATION APPLICATIONS 2 - MEAN AND ROOT MEAN SQUARE VALUES 13.2.1 Mean values 13.2.2 Root mean square values 13.2.3 Exercises 13.2.4 Answers to exercises (4 pages) UNIT 13.3 - INTEGRATION APLICATIONS 3 - VOLUMES OF REVOLUTION 13.3.1 Volumes of revolution about the x-axis 13.3.2 Volumes of revolution about the y-axis 13.3.3 Exercises 13.3.4 Answers to exercises (7 pages) UNIT 13.4 - INTEGRATION APPLICATIONS 4 - LENGTHS OF CURVES 13.4.1 The standard formulae 13.4.2 Exercises 13.4.3 Answers to exercises (5 pages) UNIT 13.5 - INTEGRATION APPLICATIONS 5 - SURFACES OF REVOLUTION 13.5.1 Surfaces of revolution about the x-axis 13.5.2 Surfaces of revolution about the y-axis 13.5.3 Exercises 13.5.4 Answers to exercises (7 pages) UNIT 13.6 - INTEGRATION APPLICATIONS 6 - FIRST MOMENTS OF AN ARC 13.6.1 Introduction 13.6.2 First moment of an arc about the y-axis 13.6.3 First moment of an arc about the x-axis 13.6.4 The centroid of an arc 13.6.5 Exercises 13.6.6 Answers to exercises (11 pages) 12 of 20 UNIT 13.7 - INTEGRATION APPLICATIONS 7 - FIRST MOMENTS OF AN AREA 13.7.1 Introduction 13.7.2 First moment of an area about the y-axis 13.7.3 First moment of an area about the x-axis 13.7.4 The centroid of an area 13.7.5 Exercises 13.7.6 Answers to exercises (12 pages) UNIT 13.8 - INTEGRATION APPLICATIONS 8 - FIRST MOMENTS OF A VOLUME 13.8.1 Introduction 13.8.2 Firs...
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