Newton-Leibniz formula, also known as the basic theorem of calculus, reveals the relationship between definite integral and original function or indefinite integral of integrand function.
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.
1670, isaac barrow, a British mathematician, said in his book "Lecture Notes on Geometry" that the tangent problem is an inverse proposition of the area problem in geometric form, and it is actually a geometric expression of Newton-Leibniz formula.
1666 10 Newton solved the problem of how to solve the displacement of an object according to its velocity in his first calculus paper, and discussed how to solve the area enclosed by a curve according to this operation, and put forward the basic theorem of calculus for the first time.
The German mathematician Leibniz found that the area of a curve depends on the sum of the vertical coordinates between infinite cells. 1677. In a manuscript, Leibniz clearly expounded the basic theorem of calculus: given a curve whose ordinate is y, if there is a curve Z that makes dz/dx=y, the area under the curve Y is ∫ YDX = ∫.
The discovery of Newton-Leibniz formula makes people find a general method to solve the problems of curve length, area enclosed by curve and volume enclosed by surface. The calculation of definite integral is simplified. As long as we know the original function of the integrand, we can always find the exact value of the definite integral or an approximate value with certain precision.