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The first fundamental theorem of calculus tell us that we can always construct a primitive of a continuous function by integration. When we combine this with the fact that two primitives of the same function can differ only by a constant, we obtain the Second Fundamental Theorem of Calculus. (Apostol)

The Second Fundamental Theorem of Calculus (with a weak hypothesis) says: Assume f is continuous on an open interval I, and let P be any primitive (an indefinite integral, P'=f) of f on I. Then, for each a and each b in I, we have

Fundamental Theorem of Calculus | matematicasVisuales

The demonstration is not difficult:

Let

Fundamental Theorem of Calculus | matematicasVisuales

Then, for the First Theorem of Calculus:

There is a constant C such that

We can evaluate C because

Then C is

We can write

This expression is true for x=b, and our result follows:

This theorem tells us that we can compute the value of a definite integral by a mere subtraction if we know a primitive (antiderivative) F. The problem of evaluating an integral is transferred to another problem, that of finding a primitive F of f. Every differentiation formula, when read in reverse, gives us an example of a primitive of some function f and this, in turn, leads to an integration formula for this function. (Apostol)

We want to calculate a definite integral of a function f:

Fundamental Theorem of Calculus: a function and a definite integral | matematicasVisuales

The integral function F is:

Fundamental Theorem of Calculus: a function and the integral function | matematicasVisuales

If we know how to calculate another primitive or antiderivative P of f we can calculate very easily (only substracting) the value of the integral:

Fundamental Theorem of Calculus: calculating an integral substracting two values of an antiderivative | matematicasVisuales

If we choose another antiderivative, the result is the same:

Fundamental Theorem of Calculus: using different antiderivatives the value of the integral is the same | matematicasVisuales

Remember how clever Archimedes was when he calculated the area of a parabolic segment:

Fundamental Theorem of Calculus: Method of Archimedes to calculate the area of a parabolic segment | matematicasVisuales

Or the different techniques that Cavalieri, Fermat and others needed to integrate power functions (1800 years later):

Fundamental Theorem of Calculus: Fermat and other knew how to calculate integrals of power functions | matematicasVisuales

With the powerful tool that this theorem provided us, to integrate this kind of integrals is routine. For example, if we want to integrate

Fundamental Theorem of Calculus | matematicasVisuales

We look for a primitive or antiderivative of the integrand:

And we apply the theorem:

The result is:

And, in general, it is easy to integrate power functions:

The Second Fundamental Theorem of Calculus provides us with a powerful tool for evaluating definite integrals exactly but is useful only when we can fin an antiderivative for the function being integrated. Sometimes this is a easy task but sometimes it is difficult. In order to use the theorem in the evaluation of definite integrals we must develop some procedures to aid in finding antiderivatives. They are called 'Techniques of Integration'.

It is very typical to use the letter F for a primitive or antiderivative of f. And to denote the difference F(b)-F(a), as a short-hand notation, we use the symbol

Fundamental Theorem of Calculus | matematicasVisuales

A basic example:

Another basic example: we know that Archimedes was able to calculate the area of a parabolic segment. Now we can use the Fundamental Theorem of Calculus. We want to calculate this area:

Fundamental Theorem of Calculus: parabolic segment | matematicasVisuales

First of all we need to find the equation of the parabola (a second degree polynomial):

We are going to evaluate the area using the integral:

And the calculation is straightforward:

As I said before, the hypothesis of the Second Fundamental Theorem is stronger because it applies to every integrable function f (not necessarily continuous). This is a more difficult result to probe.

REFERENCES

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.

MORE LINKS

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Polynomial functions and integral (1): Linear functions
It is easy to calculate the area under a straight line. This is the first example of integration that allows us to understand the idea and to introduce several basic concepts: integral as area, limits of integration, positive and negative areas.
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To calculate the area under a parabola is more difficult than to calculate the area under a linear function. We show how to approximate this area using rectangles and that the integral function of a polynomial of degree 2 is a polynomial of degree 3.
Archimedes' Method to calculate the area of a parabolic segment
Archimedes show us in 'The Method' how to use the lever law to discover the area of a parabolic segment.
Integral of powers with natural exponent
The integral of power functions was know by Cavalieri from n=1 to n=9. Fermat was able to solve this problem using geometric progressions.
Polynomial functions and derivative (5): Antidifferentiation
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Definite integral
The integral concept is associate to the concept of area. We began considering the area limited by the graph of a function and the x-axis between two vertical lines.
Monotonic functions are integrable
Monotonic functions in a closed interval are integrable. In these cases we can bound the error we make when approximating the integral using rectangles.
Indefinite integral
If we consider the lower limit of integration a as fixed and if we can calculate the integral for different values of the upper limit of integration b then we can define a new function: an indefinite integral of f.
Polynomial functions and derivative (1): Linear functions
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The derivative of a cubic function is a quadratic function, a parabola.
Polynomial functions and derivative (4): Lagrange polynomials (General polynomial functions)
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Powers with natural exponents (and positive rational exponents)
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Polynomial Functions (3): Cubic functions
Polynomials of degree 3 are cubic functions. A real cubic function always crosses the x-axis at least once.
Polynomial Functions (4): Lagrange interpolating polynomial
We can consider the polynomial function that passes through a series of points of the plane. This is an interpolation problem that is solved here using the Lagrange interpolating polynomial.
Archimedes and the area of an ellipse: an intuitive approach
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Archimedes and the area of an ellipse: Demonstration
In his book 'On Conoids and Spheroids', Archimedes calculated the area of an ellipse. It si a good example of a rigorous proof using a double reductio ad absurdum.
Kepler: The Area of a Circle
Kepler used an intuitive infinitesimal approach to calculate the area of a circle.
Kepler: Surface and volume of a sphere
Kepler studied the volume and surface of the sphere. He thought the volume of the sphere as made up of small cones, then he sum all of these cones and get a relation between the surface of a sphere en its volume.
Kepler: The volume of a wine barrel
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Kepler: The best proportions for a wine barrel
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Cavalieri: The volume of a sphere
Using Cavalieri's Principle we can calculate the volume of a sphere.