In the wonderful world of mathematics, integral science occupies a pivotal position. It is not only a bridge between differential calculus and practical applications, but also a central part of analytics. Today, we will demystify the two major concepts of the Riemann integral and the definite integral, and explore the relationship and differences between them.
1. The birth and definition of the Riemann integral
The Riemann integral, as the starting point of integralism, was first proposed by the German mathematician Riemann. It is based on the idea of an intuitive approximate summation, which divides the definition domain of a function into an infinite number of small rectangles, and then uses the sum of the areas of these small rectangles to approximate the integral value of the function.
Specifically, given a closed interval [a, b] and a bounded function f(x) on that interval, we can split [a, b] into n intervals [x, x i], where i is from 1 to n. Select a representative point x i * on each interval, and then calculate the product of f(x i *) and the width of the interval, i.e., f(x i *) x i - x). Finally, adding these products gives us a Riemann sum of f(x) on [a, b]. As the partitions become finer and finer, i.e., n becomes larger and larger, the Riemannian sum will approach a stable value, which is the Riemann integral of f(x) on [a, b].
2. The concept and properties of definite integrals
Unlike the Riemann integral, the definite integral is a more general concept of integration based on the theory of measures. It is not only suitable for bounded functions, but also for functions that cannot be handled in the framework of the Riemann integral, such as some discontinuous functions.
The definition of definite integrals relies on the so-called Lebegus measure. In simple terms, for a set a, the Lebegus measure the size of the space that A "occupies". In the context of a definite integral, this "space" is a line segment on the real number axis. Thus, a definite integral can be seen as a measure of the area "occupied" by the value of a function on the real number axis.
3. The relationship between Riemann integrals and definite integrals
In practical applications, many functions are continuous, which means that their Riemann integrals are equal to definite integrals. This is because continuous functions are derivable almost everywhere on the real number line, so that their images are almost visually smooth, with no jumps or breakpoints. In this case, whether the Riemann sum is used to approximate or the Lebeig measure is used, the results obtained are consistent.
However, there are also functions that, although bounded, have discontinuities or anomalous behavior at certain points, such as the dirichlet function. Such a function is non-integrable in the framework of the Riemann integral, because no matter how we divide the interval and choose the representative points, the Riemannian sum cannot stably approximate a value. However, in the framework of definite integrals, we can still define integral values for these functions, because definite integrals consider the overall distribution of function values on the real number axis, not just local behavior.
IV. Conclusion
As the two cornerstones of integralism, Riemann integral and definite integral have their own unique charm and application value. By gaining a deeper understanding of their definitions, properties, and relationships, we will not only be able to better understand the nature of integrals, but also be able to flexibly apply these tools to solve complex mathematical problems in real-world problems. Hopefully, this article will spark your interest in integral science and lead you into this challenging and fun world of mathematics.
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