newton leibniz formula

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.

Second, a brief history of development

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.

Third, application

Newton-Leibniz formula simplifies the calculation of definite integral, which can be used to calculate the arc length of curve, the area enclosed by plane curve and the solid volume enclosed by space surface. It is widely used in practical problems, such as calculating the filling volume of dam body.

Newton-Leibniz formula is also widely used in physics, calculating the distance of moving objects, calculating the work done by variable force along a straight line and the attraction between objects. Newton-Leibniz formula promotes the development of other branches of mathematics, which are embodied in differential equations, Fourier transform, probability theory, complex variable functions and other branches of mathematics.

Newton-Leibniz formula is the pillar of integral theory. Newton-Leibniz formula can be used to prove the definite integral substitution formula, the first mean value theorem of integral and Taylor formula of integral remainder. Newton-Leibniz formula can also be extended to double integral and curve integral, from one dimension to multiple dimensions.