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 mathematical concept of calculus is a branch of mathematics that studies the differential and integral of functions and related concepts and applications in higher mathematics. It is the basic subject of mathematics.
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. 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.
Theorem significance:
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. Newton-Leibniz formula is a bridge between differential calculus and integral calculus, and it is one of the most basic formulas in calculus.
It is proved that differential and integral are reversible operations, and it marks that calculus has formed a complete system in theory, and calculus has since become a real discipline. 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.