Note: The formal name of definite integral is Riemann integral. In Riemann's own words, the image of a function in a rectangular coordinate system is divided into countless rectangles by a straight line parallel to the Y axis, and then the rectangles in a certain interval [a, b] are accumulated to get the image area of this function in the interval [a, b]. In fact, the upper and lower limits of definite integral are the two endpoints A and B of the interval.

Extended data:

General theorem

Theorem 1: If f(x) is continuous in the interval [a, b], then f(x) is integrable in [a, b].

Theorem 2: If the interval f(x) is bounded on [a, b] and there are only finite discontinuous points, then f(x) is integrable on [a, b].

Theorem 3: Let f(x) be monotone in the interval [a, b], then f(x) can be integrated in [a, b].

Newton-Leibniz formula

Definite integral and indefinite integral seem to have nothing to do, but they are closely related in essence because of the support of a mathematically important theory. It seems impossible to subdivide a graph infinitely and then accumulate it, but because of this theory, it can be transformed into calculating integral. This important theory is the famous Newton-Leibniz formula, and its content is:

If f(x) is a continuous function on [a, b] and f'(x)= f(x), then

Expressed in words: the value of the definite integral formula is the difference between the value of the original function at the upper limit and the value of the original function at the lower limit.

It is precisely because of this theory that the relationship between integral and Riemann integral is revealed, which shows its important position in calculus and even higher mathematics. Therefore, Newton-Leibniz formula is also called the basic theorem of calculus.

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