Historical Reflections On Teaching the

Fundamental Theorem Of Integral Calculus

David Bressoud Macalester College St. Paul, MN

IllinoisSec+onAugustanaCollegeRockIsland,ILApril9,2010

PowerPointavailableatwww.macalester.edu/~bressoud/talks

MAA

“The task of the educator is to make the child’s spirit pass again where its forefathers have gone, moving rapidly through certain stages but suppressing none of them. In this regard, the history of science must be our guide.”

Henri Poincaré

1854–1912

The history of mathematics informs teaching by

•  helping students understand the process of mathematical discover,

•  explaining the motivation behind definitions and assumptions, and

•  illuminating conceptual difficulties.

Series, continuity, differentiation

1800–1850

Integration, structure of the real numbers

1850–1910

WhatistheFundamentalTheoremofCalculus?

Whyisitfundamental?

Possibleanswer:Integra+onanddifferen+a+onareinverseprocessesofeachother.

Problemwiththisanswer:Formoststudents,theworkingdefini+onofintegra+onistheinverseprocessofdifferen+a+on.

For any function f that is continuous on the interval [a,b],

and, if F’(x) = f(x) for all x in [a,b], then

ddx

f t( )dt = f x( ),a

x

∫

f t( )dt = F b( ) − F a( )a

b

∫ .

Differen+a+onundoesintegra+on

Definiteintegralprovidesanan+deriva+ve

Integra+onundoesdifferen+a+on

Anan+deriva+veprovidesameansofevalua+ngadefiniteintegral.

The key to understanding this theorem is to know and appreciate that the definite integral is a limit of Riemann sums:

f x( )a

b

∫ dx is defined to be a limit over all partitions of [a,b],

P = a = x0 < x1 < < xn = b{ },

and all choices of tags: xi* ∈ xi−1, xi[ ], of

f x( )a

b

∫ dx = limP→0

f xi*( ) xi − xi−1( )

i=1

n

∑ ,

where P = maxi

xi − xi−1( ).

Riemann’shabilita-onof1854:

ÜberdieDarstellbarkeiteinerFunc-ondurcheinetrigonometrischeReihe

Purpose of Riemann integral:

1.  Toinves+gatehowdiscon+nuousafunc+oncanbeands+llbeintegrable.Canbediscon+nuousonadensesetofpoints.

2.  Toinves+gatewhenanunboundedfunc+oncans+llbeintegrable.Introduceimproperintegral.

limmax Δxi→0

f xi*( )

i=1

n

∑ Δxi1826–1866

Riemann’s function: f x( ) = nx{ }n2n=1

∞

∑

x{ } =x − nearest integer( ), when this is < 1

2,

0, when distance to nearest integer is 12.

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At x = a 2b , gcd a,2b( ) = 1, the value of the function jumps by π2

8b2 .

Riemann’s definition of the definite integral is not widely adopted by mathematicians until the 1870s.

First appearance of the Fundamentalsatz der Integralrechnumg (Fundamental Theorem of Integral Calculus) in its modern form was in an appendix to a paper on Fourier series by Paul du Bois-Reymond in 1876.

1831–1889

This definition was used as recently as Granville’s Elements of the differential and integral calculus, the dominant American textbook of the first half of the 20th century.

Before Cauchy, the definite integral was defined as the difference of the values of an antiderivative at the endpoints.

FUNDAMENTALTHEOREMOFTHEINTEGRALCALCULUS.Let ϕ(x) be continuous for the interval x = a to x = b. Let this interval be divided into n subintervals whose lengths are and points be chosen, one in each subinterval their abscissas being respectively. Consider the sum

(C)

Then the limiting value of this sum when n increases without limit equals the value of the definite integral [Granville,2ndedi+on(1911),p.363]

Thisstatementofthefundamentaltheoremdidnotchangethroughalledi+ons(lastwasthe6thin1946).

Δx1,Δx2 ,…,Δxn

x1, x2 ,…, xn

φ x1( )Δx1 + φ x2( )Δx2 ++ φ xn( )Δxn = φ xi( )Δxii=1

n

∑ ,

Cauchy, 1823, first explicit definition of definite integral as limit of sum of products

Purpose is to show that the definite integral is well-defined for any continuous function.

f x( )dx = limn→∞a

b

∫ f xi−1( )i=1

n

∑ xi − xi−1( ).

1789–1857

He now needs to connect this to the antiderivative. Using the mean value theorem for integrals, he proves that

ddt

f x( )0

t

∫ dx = f t( ).

1789–1857

By the mean value theorem, any function whose derivative is 0 must be constant. Therefore, any two functions with the same derivative differ by a constant. Therefore, if F is any antiderivative of f, then

f x( )0

t

∫ dx = F t( ) + C ⇒ f x( )a

b

∫ dx = F b( ) − F a( ).

18th century: Those using calculus are well aware of the connection between antiderivatives and limits of sums of the form but this connection almost never appears explicitly in the literature.

f xi−1( )i=1

n

∑ xi − xi−1( ),

The standard calculus text of the first half of the 19th century was Lacroix’s Traité élémentaire du calcul différentiel et du calcul intégral, first published in 1802. No explicit mention is made of this connection until the 7th (1867) edition when it was added by Joseph Alfred Serret.

1646–1716

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“I shall now show that the general problem of quadratures [areas] can be reduced to the finding of a line that has a given law of tangency (declivitas), that is, for which the sides of the characteristic triangle have a given mutual relation. Then I shall show how this line can be described by a motion that I have invented.”

Supplementum geometriae dimensoriae, Acta Eruditorum, 1693

Leibniz seeks to find the area under the curve AH. If we can find a curve whose slope is proportion to the ordinate FH, then the change in the distance to this curve is equal to the area.

1646–1716

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The characteristic triangle is TBC.

The differential triangle is (C)EC with base = dx and height = dy.

dy :dx = TB :BC = FH :a, where a is a fixed length

FH dx = ady

Summing the quantities FH dx produces the area under the curve AH. Summing the quantities a dy produces a times the change in y. Therefore, the change in y is proportional to the accumulated area.

1630–1677

Isaac Barrow, 1670, Lectiones geometricae, demonstrates that if we are given a curve and construct a second curve so that its ordinate is proportional to the accumulated area under the first curve, then the slope of the second curve is equal to the ordinate of the first.

1638–1675

James Gregory, 1668, Geometriae pars universalis, proves a statement equivalent to Barrow’s result, but it is hidden inside his treatment of the arc length formula.

1643–1727

Isaac Newton, the October 1666 Tract on Fluxions (unpublished):

“Problem 5: To find the nature of the crooked line [curve] whose area is expressed by any given equation.”

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Let y be the area under the curve ac, then “the motion by which y increaseth will be bc = q.”

Newton does not prove this result. He simply observes that the ordinate expresses the rate at which the area is changing.

1643–1727

Isaac Newton, the October 1666 Tract on Fluxions (unpublished):

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We are given a formula for q. From problem 5, if we can find a curve whose rate of change is given by q, then the change in that curve describes the change in area.

“Problem 7: The nature of any crooked line being given to find its area, when it may bee.”

To find area under y = axa2 − x2

,

observe that the fluxion (derivative) of2cnb

a + bxn is cxn

x a + bxn.

1877–1947

G.H. Hardy A Course of Pure Mathematics, 1908:

“The ordinate of the curve is the derivative of the area, and the area is the integral of the ordinate.”

The height of a curve is the rate at which the area under that curve is changing.

If the height of a curve describes the rate at which some quantity is accumulating, then the area under the curve represents the total amount that has accumulated.

1323–1382

Nicole Oresme Tractatus de configurationaibus qualitatum et motuum (Treatise on the Configuration of Qualities and Motions) Circa 1350

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Geometric demonstration that, under uniform acceleration, the distance traveled is equal to the distance traveled at constant average velocity.

1588–1637

Isaac Beeckman’s journal, 1618, unpublished in his lifetime

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Beeckman gives a justification, using a limit argument, that if the ordinate of a curve represents velocity, then the accumulated area under that curve represents the distance traveled.

Three Lessons: 1. The real point of the FTIC is that there are two conceptually different but generally equivalent ways of interpreting integration: as antidifferentiation and as a limit of approximating sums.

2. The modern statement of the FTIC is the result of centuries of refinement of the original understanding and requires considerable unpacking if students are to understand and appreciate it.

3. The FTIC arose from a dynamical understanding of total change as an accumulation of small changes proportional to the instantaneous rate of change. This is where we need to begin to develop student understanding of the FTIC.

PowerPointavailableatwww.macalester.edu/~bressoud/talks