Represented by the symbol ∫, an integral is the result of performing integration. It is the area, or a generalization of area, under any section of a graph that is described by a function f(x); in other words, the continuous cumulative sum, ∫f(x)dx, of a function. A definite integral has a numerical value which represents the area bounded by the x-axis, y = f(x) and the two lines x = a and y = b. If a and b are unspecified the integral is said to be indefinite. In many cases, the derivative of the indefinite integral of a function gives the original function itself, so integration can be regarded as the reverse of differentiation.
Not all functions have an exact formula that allows an integral to be found. In such cases, numerical integration has to be used, in which the area is found using approximate numerical techniques. Integrals, together with derivatives, are the fundamental objects of calculus.
Related category CALCULUS AND ANALYSIS
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