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:
- The value of the defined integral changes sign if the integration limits are permuted.
- If the limits whose integration coincide, the defined integral is zero.
- 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].
- The definite integral of a sum of functions is equal to the sum of integrals ·
- 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.