Top-down Calculus

Chapter 4

Integrals

S. Gill Williamson

Gill Williamson Home Page

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Preface

This chapter, Chapter 4 of Top-down Calculus, is devoted to the basic conceptsand techniques of integral calculus. Software systems have been developedto compute integrals and express the answers in terms of standard functions(when possible). Some of these products are available for free on the webor for low cost to students. As with differentiation, these programs are nosubstitute for understanding what your are doing.

Top-down Calculus was developed in the 1980’s for a summer session programto train high school teachers in San Diego County to teach calculus. Theseteachers had all taken calculus themselves, but they were wary of standingbefore a class and fielding questions – math anxiety of the ”second kind.”They knew about Newton, Leibniz, the falling apple, etc. What they didn’tknow was how to respond quickly when a student asked, ”Hey teacher, howdo I work this one?”

My approach in this book is to emphasize intuition and technique. The im-portant chain rule is presented intuitively on page 11 (instead of page 100+ asin many standard calculus books). Exercises are presented as follows: proto-typical exercise set, solutions and discussion, numerous exercise sets that arevariations on the prototypes. My students (the teachers) were encouraged towork one or two of the variation exercise sets in detail and then scan the re-maining variations noting the techniques required for each problem. The ideawas that this “scanning” process would prepare them to deal with their ownstudents’ questions. They would be able to say with some confidence, “Well,Johnny, why don’t you try this approach.” At least this would buy time forthem to think about the question more carefully.

Subsequent to the summer session program for high school teachers, I usedthis material for the one-quarter calculus course that I regularly taught in theDepartment of Mathematics at the University of California, San Diego. Itseemed to work well for that purpose, but it is no competitor for the magnifi-cent (but very expensive) standard calculus books. I also used this material forcalculus taught in summer session. There, the concise nature of this materialworked very well.

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Table of Contents Chapter 4

Given f (x), find F(x) such that F ′(x) = f (x) .........................150

Linearity of the integral...........................................................152

Chain rule in reverse................................................................153

Differential notation.................................................................156

Exercises 4.5..............................................................................158

Variations on Exercises 4.5 ........................................... 159 - 161

Fundamental theorem of calculus .........................................162

Riemann sum ...........................................................................166

Exercises 4.15............................................................................168

Solutions to Exercises 4.15 ......................................................169

Variations on Exercises 4.15.......................................... 175 - 177

Integration by parts .................................................................178

Trigonometric integrals ...........................................................181

Trigonometric substitution......................................................182

Variations on integration exercises.........................................184

Partial fractions........................................................................191

Arclength, surface areas, volumes ..........................................200

Exercises 4.48............................................................................200

Solutions to Exercise 4.48 ........................................................201

Variations on Exercises 4.48.......................................... 208 - 211

Double integrals ......................................................................211

Volume under a surface...........................................................212

Two dimensional Riemann sum..............................................214

Iterated integrals, polar coordinates ........................... 215 - 218

Integrals in three dimensions .................................................219

Cylindrical, spherical coordinates .........................................222

Exercises 4.71............................................................................225

Index...................................................................................Index 1

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arclength, 201, 204, 206area, 168chain rule in reverse, 153

integral, 149integration by parts, 178

differential notation, 155

limacon, 174

fundamental theorem of calculus, 162, 164

parametric equation, 173partial fractions, 191, 193polar coordinates, 169

parabola, 171linearity of integral, 152limits of integration, 166, 216, 220

trigonometric integrals, 181

spherical coordinates, 222, 223

Riemann sum, 166, 167, 214signed area function, 162, 163

product rule in reverse, 178rational function, 191

trigonometric substitution, 182triple integrals, 219volume by cross-section, 201volume element, 220volume in polar coordinates, 207

volume under a surface, 212

INDEX Chapter 4

antiderivative, 149, 151

double integrals, 211 - 218

surface area, 201, 205, 206

volume of revolution, 201, 202, 203

Index 1