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We have already seen how to define a definite integral.

Suppose now that f is integrable on [a,b]. We shall keep a and f fixed, then can define a new function on [a,b] by

Indefinite integral: formula | matematicasVisuales

This is called an indefinite integral.

If f is positive, F(x) is sometimes called an Area function.

Indefinite integral: Area function | matematicasVisuales

We say an indefinite integral rather than the indefinite integral because F also depends on the lower limit a. Different values of a will lead to different functions F. But the difference between two integral functions of the same function is independent of x, they differ only by a constant. [Apostol]

We can see a very similar behavior when we study the antiderivative concept.

If f is positive in an interval, then F (in this case F is area) is increasing.

Indefinite integral: positive function, increasing integral | matematicasVisuales

If f is negative in an interval, then F is decreasing.

Indefinite integral: negative function, decreasing integral | matematicasVisuales

If f(x)=0 then x is a critical point of F.

Indefinite integral: critical points of the integral | matematicasVisuales
Indefinite integral: critical points of the integral | matematicasVisuales

This three relationships between F and f are precisely those enjoyed by a function and its derivative.

We can start studying integrals using simple polynomial functions: linear, quadratic and general polynomial functions.


Michael Spivak, Calculus, Third Edition, Publish-or-Perish, Inc.
Tom M. Apostol, Calculus, Second Edition, John Willey and Sons, Inc.


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