Definite integral

Definite integral . Given a function f (x) and an interval [a, b], the definite integral is equal to the limited area between the graph of f (x), the abscissa axis, and the vertical lines x = a and x = b.


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  • 1 Definition
  • 2 Properties
  • 3 Example
  • 4 Applications
  • 5 See also
  • 6 Sources


The definite integral is one of the fundamental concepts of Mathematical Analysis .

The definite integral of f (x) on the interval [a, b] is equal to the limited area between the graph of f (x) , the axis of abscissa, and the vertical lines x = a and x = b (under the hypothesis that the function f is positive). This integral is represented by:

a is the lower limit of integration and b is the upper limit of integration.

If the function F is a primitive function of f on the interval [a, b], by Barrow’s Rule we have to:


  1. The value of the defined integral changes sign if the integration limits are permuted.
  2. If the limits whose integration coincide, the defined integral is zero.
  3. If c is an interior point of the interval [a, b], the definite integral decomposes as a sum of two integrals extended at the intervals [a, c] and [c, b].
  4. The definite integral of a sum of functions is equal to the sum of integrals ·
  5. The integral of the product of a constant times a function is equal to the constant times the integral of the function.


The example image has an error in the first term, when integrating x ^ 2 it remains x ^ 3/3, then when replacing 2 in x, it remains 8/3, being the result 2/3


The concept of integral had its historical origin in the need to solve specific problems such as: calculation of area limited by two curves, arc lengths , volumes , work , speed , moments of inertia , etc .; All of these calculations can be performed using the defined integral.


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