The content of Newton-Leibniz formula is that the definite integral of a continuous function in the interval [a, b] is equal to the increment of any of its original functions in the interval [a, b]. Newton described this formula with kinematics in the Introduction to Flow Number written by 1666, and Leibniz formally proposed this formula in a manuscript written by 1677. Because they first discovered this formula, they named it Newton-Leibniz formula.

Newton-Leibniz formula provides an effective and simple calculation method for a given integral, which greatly simplifies the calculation process of the definite integral.

General theorem of definite integral

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].

Extended data

The formal name of definite integral is Riemann integral. It is to divide the image of a function in a rectangular coordinate system into countless rectangles with a straight line parallel to the Y axis, and then add the rectangles in a certain interval [a, b] to get the area of the image 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.

The definite integral is to divide the image [a, b] of a function in a certain interval into n parts, divide it into countless rectangles with a straight line parallel to the Y axis, and then find the sum of all these rectangular areas when n→+∞. Traditionally, we use arithmetic progression to divide points, that is, the distance between adjacent ends is equal. But it must be pointed out that even if they are not equal, the integral value is still the same.

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