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How is the definite integral formula derived?
Elementary definite integral is to calculate the large area under the curve. Methods The interval of the posterior product variable is divided into infinitesimal units, multiplied by the response function value to approximate sum and take the limit, which can prove that if the integral variable is an independent variable, the integral and derivative operations are inverse operations. (Newton Leibniz formula)

Integration is the inverse operation of differentiation, that is, knowing the derivative function of a function and finding the original function in reverse. In application, the function of integration is not only this, but also widely used in summation, that is, to find the area of curved triangle. This ingenious solution is determined by the special properties of integral.

A function can have indefinite integral, but not definite integral; There can also be definite integral, but there is no indefinite integral. A continuous function must have definite integral and indefinite integral; If there are only a finite number of discontinuous points, the definite integral exists; If there is jump discontinuity, the original function must not exist, that is, the indefinite integral must not exist.

Extended data:

Let λ = λ=max{△x 1, △x2, …, △xn} x2, ..., △ xn} (that is, λ is the maximum interval length). If λ→0, the limit of integral sum exists, then this limit is called the definite integral of function f(x) in the interval [a, b] and the definite integral of function f(x) in the interval [a, b].

The integrand function does not necessarily have only one variable, and the integral domain can also be a space with different dimensions, even an abstract space without intuitive geometric significance.

Let f(x) be continuous in the interval [a, b], then f(x) can be integrated in [a, b]. Let the interval f(x) be bounded on [a, b] and have only a finite number of discontinuous points, then f(x) can be integrated on [a, b]. Let f(x) be monotonic in the interval [a, b], then f(x) can be integrated in [a, b].

Baidu encyclopedia-definite integral