Mathematical Time Capsules: Integration à la Fermat

Move over Riemann and make room for Fermat! Most textbooks on the integral calculus focus heavily on the Riemann integral when introducing integration. This method is very effective in transitioning students from the finite (or macro) world of finding area geometrically to the infinite (or micro) world of finding area by integration. Once the notation and abstract idea of an area made up of an infinite number of infinitely thin slices is mastered, most textbooks move directly on to integration techniques. Finding areas using rectangles is usually not mentioned again except in review, to help students visualize a more difficult example, or when transitioning to finding volumes using double integrals.

Riemann used rectangles of uniform width. This is very handy when letting the width, dx, tend to zero. It also corresponds nicely to the definition of the derivative presented in most textbooks, in which the width h = xi+1 – xi in the denominator approaches 0. I still advocate introducing integration in this manner. However, there is no reason to stop there and move directly on to integration techniques.

Prior to Riemann, even prior to Newton and Leibniz, Fermat and others were finding areas using the sum of thin rectangles. However, Fermat's rectangles were not of uniform width. The width of Fermat's rectangles decreased based on a geometric series. Looking at Fermat's method directly after introducing the Riemann integral broadens the student's perspective on the integral calculus. Also, his techniques can be presented in an analysis course to provide depth to the material.

Frontmatter: Read PDF	pp. i-vi
Preface: Read PDF	pp. vii-x
Contents: Read PDF	pp. xi-xviii
1 - The Sources of Algebra: Read PDF By Roger Cooke	pp. 1-6
2 - How to Measure the Earth: Read PDF By Lawrence D'Antonio	pp. 7-16
3 - Numerical solution of equations: Read PDF By Roger Cooke	pp. 17-22
4 - Completing the Square through the Millennia: Read PDF By Dick Jardine	pp. 23-28
5 - Adapting the Medieval “Rule of Double False Position” to the Modern Classroom: Read PDF By Randy K. Schwartz	pp. 29-38
6 - Complex Numbers, Cubic Equations, and Sixteenth-Century Italy: Read PDF By Daniel J. Curtin	pp. 39-44
7 - Shearing with Euclid: Read PDF By Davida Fischman and Shawnee McMurran	pp. 45-54
8 - The Mathematics of Measuring Time: Read PDF By Kim Plofker	pp. 55-62
9 - Clear Sailing with Trigonometry: Read PDF By Glen Van Brummelen	pp. 63-72
10 - Copernican Trigonometry: Read PDF By Victor J. Katz	pp. 73-88
11 - Cusps: Horns and Beaks: Read PDF By Robert E. Bradley	pp. 89-100
12 - The Latitude of Forms, Area, and Velocity: Read PDF By Daniel J. Curtin	pp. 101-106
13 - Descartes' Approach to Tangents: Read PDF By Daniel J. Curtin	pp. 107-110
14 - Integration à la Fermat: Read PDF By Amy Shell-Gellasch	pp. 111-116
15 - Sharing the Fun: Student Presentations: Read PDF By Amy Shell-Gellasch and Dick Jardine	pp. 117-122
16 - Digging up History on the Internet: Discovery Worksheets: Read PDF By Betty Mayfield	pp. 123-126
17 - Newton vs. Leibniz in One Hour!: Read PDF By Betty Mayfield	pp. 127-132
18 - Connections between Newton, Leibniz, and Calculus I: Read PDF By Andrew B. Perry	pp. 133-138
19 - A Different Sort of Calculus Debate: Read PDF By Vicky Williams Klima	pp. 139-150
20 - A ‘Symbolic’ History of the Derivative: Read PDF By Clemency Montelle	pp. 151-158
21 - Leibniz's Calculus (Real Retro Calc.): Read PDF By Robert Rogers	pp. 159-168
22 - An “Impossible” Problem, Courtesy of Leonhard Euler: Read PDF By Homer S. White	pp. 169-178
23 - Multiple Representations of Functions in the History of Mathematics: Read PDF By Robert Rogers	pp. 179-188
24 - The Unity of all Science: Karl Pearson, the Mean and the Standard Deviation: Read PDF By Joe Albree	pp. 189-198
25 - Finding the Greatest Common Divisor: Read PDF By J.J. Tattersall	pp. 199-202
26 - Two-Way Numbers and an Alternate Technique for Multiplying Two Numbers: Read PDF By J.J. Tattersall	pp. 203-208
27 - The Origins of Integrating Factors: Read PDF By Dick Jardine	pp. 209-214
28 - Euler's Method in Euler's Words: Read PDF By Dick Jardine	pp. 215-222
29 - Newton's Differential Equation ẏ/ẋ = 1 − 3x + y + xx + xy: Read PDF By Hüseyin Koçak	pp. 223-228
30 - Roots, Rocks, and Newton-Raphson Algorithms for Approximating √2 3000 Years Apart: Read PDF By Clemency Montelle	pp. 229-240
31 - Plimpton 322: The Pythagorean Theorem, More than a Thousand Years before Pythagoras: Read PDF By Daniel E. Otero	pp. 241-250
32 - Thomas Harriot's Pythagorean Triples: Could He List Them All?: Read PDF By Janet L. Beery	pp. 251-260
33 - Amo, Amas, Amat! What's the sum of that?: Read PDF By Clemency Montelle	pp. 261-268
34 - The Harmonic Series: A Primer: Read PDF By Adrian Rice	pp. 269-276
35 - Learning to Move with Dedekind: Read PDF By Fernando Q. Gouvêa	pp. 277-284
About the Editors: Read PDF	pp. 285-285

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