We come now to the remarkable connection that exists between integration and differentiation. The relationship between these two processes is somewhat analogous to that which holds between 'squaring' and 'taking the square root'. If we square a positive number and then take the positive square root of the result, we get the original number back again. Similarly, if we operate on a continuous function f by integration, we get a new function (an indefinite integral of f) which, when differentiated, leads back to the original function.(Apostol, pp. 202)
This connection between differentiation and integration is very surprising. Integration is related with adding together many very small numbers (for example, when we calculate an area, the length of a curve, etc.) and differentiation is the instantaneous rate of change (one graphical interpretation is the slope of the tangent to a curve). The Fundamental Theorem of Calculus tell us that this two concepts are intimately related.
We know that if f is integrable, then F(x) [an indefinite integral] is continuous; it is only fitting that we ask what happens when the original function f is continuous. It turns out that F is differentiable (and its derivative is especially simple).[Spivak]
(FUNDAMENTAL THEOREM OF CALCULUS First) Let f be an integrable function on [a,b], and define a new function F on [a,b] by
If f is continuous at c in [a,b], then F is differentiable at c, and
A well known visual demonstration assumes that the function f is continuous in some neighborhood of the point (this is a weaker condition, the hypothesis of the theorem is stronger. If you need a more analytical and rigorous proof you can read it in a good book about Calculus)
If c is in (a,b), looking at the image you could agree that
If h is small enough (to be more precise, here you can use the mean-value theorem)
Dividing by h:
If f has better properties, if f is continuous at all points in [a,b], then F is differentiable at all points in [a,b] and
The idea is, we start with a function f:
We consider an indefinite integral function F (dragging the lower limit of integration you get different integral functions):
At a point we differentiate this function F (graphically, this is the slope of the tangent line):
This FTC tell us that every continuous function has an antiderivative and shows how to construct one using an indefinite integral. Even those non-differentiable functions with "corners" such as absolute value has an antiderivative.
Sometimes the problem is how to find an antiderivative of a function: given a function f(x), find a function F(x) such that F'(x) = f(x).
An important case is when we want to integrate a function that has an antiderivative (a primitive). It is to say, you know a function f and you want to integrate f' (or you have to integrate f' and you can find an antiderivative f). In this case, we see the function to be integrate as a rate of change and the integral as an accumulator of this change (example: the integral of the velocity is the distance).
You define an integral function F (but now we are integrating f'):
And then F is and antiderivative of f', it is to say:
And you can see that
One step more and we are going to have the Second Fundamental Theorem of Calculus (or the evaluation part of the theorem).
Michael Spivak, Calculus, Third Edition, Publish-or-Perish, Inc.
Tom M. Apostol, Calculus, Second Edition, John Willey and Sons, Inc.
Otto Toeplitz, The Calculus, a genetic approach, The University of Chicago Press, 1963 (p. 95-99).
Kenneth A. Ross, Elementary Analysis: The Theory of Calculus, Springer-Verlag New York Inc., 1980 (p. 190).
Serge Lang, A First Course in Calculus, Third Edition, Addison-Wesley Publishing Company.
David M. Bressoud, Historical Reflections on Teaching the Fundamental Theorem of Calculus, American Mathematical Monthly 118 (2011).
Jorge M. López Martínez and Omar A. Hernández Rodríguez,Teaching the Fundamental Theorem of Calculus: A Historical Reflection in MathDL.