What is the integral mean value theorem? The integral mean value theorem is divided into the first integral mean value theorem and the second integral mean value theorem, and each theorem contains two formulas. Its degenerate state means that there is a moment in the process of ξ change that makes the areas of two graphs equal.
The mean value theorem of integrals reveals a method of transforming integrals into function values or complex functions into simple functions. It is a basic theorem and an important means of mathematical analysis, which is widely used in finding limits, judging some property points, estimating integral values and so on.
The generalized form of integral mean value theorem is 1. If both F and G are continuous on [a, b] and G has the same sign on [a, b], then at least one point C belongs to [a, b], so that the integral of F multiplied by G on [a, b] is equal to f(c) multiplied by G on [a, b].
2. Let the function f be integrable on [a, b]. If g is a monotone function, one point c belongs to [a, b], so that the integral of (f times g) is equal to g(a) times (the integral of f on [a, c]) plus g(b) times (the integral of f on [c, b]).
The theorem of integral mean value theorem is applied to 1 to find the limit.
In the calculation of function limit, if there is a definite integral formula, we can often use the relevant knowledge of definite integral, such as the integral mean value theorem, and apply the integral problem to some functions with integral formula, which often leads to the problem of judging whether some points with certain properties exist, and sometimes the problem can be solved by using the integral mean value theorem.
2. Use evaluation
In most integral formulas, it is rare to find the original function of the integrand and then evaluate it. When the integrand function is "non-integrable" or the original function is complex, various methods can be used to estimate the integral. For the product integrand function, estimate the slowly changing part or the part that is difficult to integrate, and integrate the integrable part. Integral mean value theorem and various inequalities are commonly used methods.
3. Proof of inequality
Integral inequality refers to an inequality that contains more than two integrals. When the integral interval is the same, the different integrals in the same integral interval are combined first, and the inequality is proved flexibly by using the integral mean value theorem according to the conditions satisfied by the integrand function.
When proving definite integral inequality, in order to get rid of the integral sign, we often consider using the integral mean value theorem. If the integrand function is the product of two functions, we can consider using the first or second integral mean value theorem. For the proof of some inequalities, we can only get the conclusion of "≥" by using the original integral mean value theorem, or we can't prove inequalities at all. After applying the improved integral mean value theorem, we can get the conclusion of ">" or solve the problem successfully.