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1 David Bressoud Macalester College St. Paul, MN Moravian College February 20, 2009 These slides will be available at

2 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

3 Series, continuity, differentiation Integration, structure of the real numbers

4 What Weierstrass Cantor did was very good. That's the way it had to be done. But whether this corresponds to what is in the depths of our consciousness is a very different question Nikolai Luzin

5 I cannot but see a stark contradiction between the intuitively clear fundamental formulas of the integral calculus and the incomparably artificial and complex work of the justification and their proofs. Nikolai Luzin

6 What is the Fundamental Theorem of Calculus? Why is it fundamental?

7 The Fundamental Theorem of Calculus (evaluation part): b If F' ( x)= f ( x), then " f! x" dx # F! b"! F! a". Differentiate then Integrate original fcn (up to constant) The Fundamental Theorem of Calculus (antiderivative part): a If f is continuous, then d dt t 0 ( ) f x dx = f ( t). Integrate then Differentiate original fcn

8 The Fundamental Theorem of Calculus (evaluation part): Differentiate then Integrate = original fcn (up to constant) The Fundamental Theorem of Calculus (antiderivative part): b If F' ( x)= f ( x), then " f! x" dx # F! b"! F! a". a If f is continuous, then d dt t 0 ( ) f x dx = f ( t). Integrate then Differentiate original fcn

9 Cauchy, 823, first explicit definition of definite integral as limit of sum of products b a f ( x)dx = lim n mentions the fact that d t dt 0 f ( x) n i= ( ) f x i dx = f ( t) en route to his definition of the indefinite integral. ( x i x i );

10 Fundamental Theorem for Integrals De L Analyse Infinitésimal, Charles de Freycinet,

11 Fundamental Theorem for Integrals De L Analyse Infinitésimal, Charles de Freycinet, Fundamental Theorem of Integral Calculus used by Paul du Bois-Reymond in appendix to paper on trigonometric series,

12 Fundamental Theorem of the Integral Calculus popularized in English by E. W. Hobson, The Theory of Functions of a Real Variable,

13 Fundamental Theorem of the Integral Calculus popularized in English by E. W. Hobson, The Theory of Functions of a Real Variable, G. H. Hardy, 2nd edition of A Course of Pure Mathematics, 94, refers to it as the Fundamental Theorem of Calculus

14 The real FTC: There are two distinct ways of viewing integration: As a limit of a sum of products (Riemann sum), As the inverse process of differentiation. The power of calculus comes precisely from their equivalence.

15 Purpose of Riemann integral: Riemann s habilitation of 854: Über die Darstellbarkeit einer Function durch eine trigonometrische Reihe lim max Δx i 0 n i= f ( * x ) i Δx i. To investigate how discontinuous a function can be and still be integrable. Can be discontinuous on a dense set of points. 2. To investigate when an unbounded function can still be integrable. Introduce improper integral.

16 Darboux s equivalent definition: Gaston Darboux Given function f bounded on [a,b] and partition P: a = x 0 < x < x 2 < < x n = b, Upper Darboux sum: Lower Darboux sum: n i= n i= ( sup f ( * x )) i ( inf f ( * x )) i Δx i Δx i f is Riemann integrable on [a,b] if and only if for each ε > 0 there is a partition P so that the difference between the upper and lower Darboux sums is less than ε: n sup f ( * x ) * ( i inf f ( x )) i Δx i < ε. i=

17 Riemann s function: f! x"#!"# x $ % nx n! " 2 n# " $ x! $ nearest integer %, when this is & 2, # %$ 0, when distance to nearest integer is 2 At x = a 2 b, gcd ( a, 2b)=, the function jumps by! 2 2 8b

18 Riemann s function: F( t) = t { nx} = 0 n= n 2 f! x"# n 2 n= $ % nx n! " 2 n# t 0 { nx} dx F is an anti-derivative. Does d dt F( t) = f ( t)? No! Darboux showed that any function that is a derivative must satisfy the intermediate value property (the function cannot have any vertical jumps). Riemann s function f has a vertical jump at every rational number with an even denominator.

19 The Fundamental Theorem of Calculus (antiderivative part): If f is continuous, then d dt t 0 ( ) f x dx = f ( t). Integrate then Differentiate original function This part of the FTC does not hold at points where f is not continuous.

20 The Fundamental Theorem of Calculus (evaluation part): b If F' ( x)= f ( x), then " f! x" dx # F! b"! F! a". a Differentiate then Integrate original fcn (up to constant) Does this work for all differentiable functions F? No! F( x) = x, 2 x df dx = 2 x is not bounded, for x > 0. and so is not Riemann integrable on [0,].

21 The Fundamental Theorem of Calculus (evaluation part): b If F' ( x)= f ( x), then " f! x" dx # F! b"! F! a". a Differentiate then Integrate original fcn (up to constant) What if the function F has a finite derivative at every point in [a,b]? No! F( x) = x 2 sin( x 2 ), df dx = 2xsin( x 2 ) 2x cos( x 2 ) is not bounded, 2xsin( x 2 ) 2x cos( x 2 ), for x 0, 0, for x = 0. and so is not Riemann integrable.

22 The Fundamental Theorem of Calculus (evaluation part): If F' ( x)= f ( x), then " f! x" dx # F! b"! F! a". What if the function F has a finite derivative at every point in [a,b] and the derivative stays bounded? b a Differentiate then Integrate original fcn (up to constant)

23 The Fundamental Theorem of Calculus (evaluation part): b If F' ( x)= f ( x), then " f! x" dx # F! b"! F! a". a Differentiate then Integrate original fcn (up to constant) Volterra, 88, constructed function with bounded derivative that is not Riemann integrable. Vito Volterra

24 " 2 x sin! " Fx x, x! 0,! "# # $ 0, x # 0.

25 " 2 x sin! " Fx x, x! 0,! "# # $ 0, x # 0. F'! x"# 2xsin! x"! cos! " x, x " 0.

26 " 2 x sin! " Fx x, x! 0,! "# # $ 0, x # 0. F'! x"# 2xsin! x"! cos! " x, x " 0. F! h"! F! 0" F'! 0"# lim h" 0 h 2 h lim sin! " # h h" 0 h # lim hsin! h"# 0. h" 0

27 " 2 x sin! " Fx x, x! 0,! "# # $ 0, x # 0. F'! x"# 2xsin! x"! cos! " x, x " 0. F! h"! F! 0" F'! 0"# lim h" 0 h 2 h lim sin! " x # h F h" 0 h # lim hsin! h"# 0. h" 0 lim! 0 ' F'! 0"! x" does not exist, but does exist (and equals 0).

28 Volterra s construction: " 2 x sin! " Start with the function Fx x, x! 0,! "# # $ 0, x # 0. Restrict to the interval [0,/8], except find the largest value of x on this interval at which F '(x) = 0, and keep F constant from this value all the way to x = /8.

29 Volterra s construction: To the right of x = /8, take the mirror image of this function: for /8 < x < /4, and outside of [0,/4], define this function to be 0. Call this function f x. ( )

30 Volterra s construction: f x ( ) is a differentiable function for all values of x, but lim f ' x and lim f ' x x! 0 " # " #! " x! 4 do not exist

31 Now we slide this function over so that the portion that is not identically 0 is in the interval [3/8,5/8].

32 f2( x) We follow the same procedure to create a new function,, that occupies the interval [0,/6] insert a copy of f2( x) into intervals of length /6 in the middle of each remaining interval.

33 We repeat this until every interval contains at least some piece of a translation of the function x 2 sin(/x). The intervals that have been filled up with copies of this function have total length n n = = = 2.

34 If we partition [0,], intervals of total length at least ½ will contain points where do not have a piece of this function. And that means that they contain an endpoint where the derivative oscillates between and +. The upper and lower Darboux sums of the derivative differ by at least 2 ½ =. The derivative of Volterra s function is not integrable!

35 The Fundamental Theorem of Calculus (evaluation part): b If F' ( x)= f ( x), then " f! x" dx # F! b"! F! a". a Differentiate then Integrate original fcn (up to constant) Henri Lebesgue To make this true, we need a better definition of the integral: Lebesgue integral. And we still need a restriction on F: Absolute Continuity.

36 Lessons:. FTC is really about connecting two very different ways of interpreting integration. Go back to calling it the Fundamental Theorem of Integral Calculus.

37 Lessons:. FTC is really about connecting two very different ways of interpreting integration. Go back to calling it the Fundamental Theorem of Integral Calculus. 2. But beware, these two ways of thinking of the integral are not entirely equivalent. This presentation is available at