The geometric significance of improper integral's existence: When there is a limit between the function and the area around the X axis, the area can be found even if the function value is infinite at one point.

For example, the geometric meaning of "X" is: the graphic area between the straight lines x=0 and x=a below the curve and above the X axis. The value of point x=a is infinite, but the area can be found.

Extended data:

The judgment of convergence and divergence of generalized integrals is essentially a question of the existence of limits and the ratio of infinitesimal or infinite. First of all, we should remember the convergence scale of two kinds of generalized integrals: for the first kind of infinite limit.

When x→+∞, f(x) must be infinitesimal, and the order of infinitesimal cannot be lower than a certain scale, so as to ensure convergence; For the unbounded function of the second kind, f(x) must be infinite when x→a+. And the order of infinitesimal can not be higher than a certain scale to ensure convergence; This scale value is generally equal to 1. Pay attention to identify improper integral.

General theorem:

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].

If f(x) is a continuous function on [a, b] and f'(x)= f(x), then

Expressed in words: the value of the definite integral formula is the difference between the value of the original function at the upper limit and the value of the original function at the lower limit.

It is precisely because of this theory that the relationship between integral and Riemann integral is revealed, which shows its important position in calculus and even higher mathematics. Therefore, Newton-Leibniz formula is also called the basic theorem of calculus.