Riemann integral

The Riemann integral is an important idea in mathematics, especially in a subject called real analysis. It was created by a mathematician named Bernhard Riemann and first shared with teachers at the University of Göttingen in 1854, though it wasn’t published until 1868. This integral gives a precise way to define what we mean by the integral of a function on an interval—a concept that helps us understand areas under curves and many other practical problems.

For many useful functions, we can find the value of the Riemann integral using something called the fundamental theorem of calculus. We can also approximate these integrals with numerical methods or even simulate them using techniques like Monte Carlo integration. Thanks to Riemann’s work, mathematicians and scientists have a strong foundation for studying and solving complex problems involving continuous functions.

Overview

The Riemann integral is a way to calculate the area under a curve on a graph. It was created by mathematician Bernhard Riemann and helps us understand how to add up tiny pieces of area to find the total area between two points.

To use the Riemann integral, we break the area into small rectangles. We then add up the areas of these rectangles. By making the rectangles thinner and thinner, we can get a very good guess at the real area. This idea is very useful in many areas of science and engineering.

Main article: Riemann sum

Similar concepts

The Riemann integral can also be defined as the Darboux integral, which is simpler and works for the same functions. Some books use special types of partitions for calculating integrals, such as "left-hand" and "right-hand" sums, where values are taken at the left or right end of each interval. These methods work fine if used carefully.

However, using both regular intervals and left-hand or right-hand sums can lead to mistakes. For example, a special function that is 1 at rational numbers and 0 elsewhere might seem to have an integral of 1 over certain intervals, but this does not match the true value. The Riemann integral avoids such problems by not integrating functions like this. The Lebesgue integral handles these cases differently, giving a value of 0 for such integrals.

Properties

The Riemann integral has a special property called linearity. This means that if you have two functions and add them together, or multiply them by numbers, the integral of the result is just the added or multiplied integrals of the original functions. In simple terms, it behaves nicely with basic math operations.

Because the integral gives a single number for a function, it acts like a straightforward tool that works well with functions that can be integrated using this method.

Integrability

A bounded function on an interval is Riemann integrable if it is continuous almost everywhere, meaning it only stops being continuous at a very small set of points. This idea was proven by mathematicians Giuseppe Vitali and Henri Lebesgue in 1907.

If a function changes at only a few points or at a countable list of points, it can still be Riemann integrable. Also, if a function is steady (monotone) on an interval, it is always Riemann integrable.

Generalizations

The Riemann integral can be extended to work with functions that have values in higher-dimensional spaces, like the space of all coordinates (Rⁿ). This extension works by integrating each coordinate separately.

The Riemann integral is normally defined only for functions on finite intervals. To use it on infinite intervals, we can define the integral as a limit of integrals over larger and larger finite intervals. However, this approach has some tricky parts and does not always match other ways of calculating integrals. For example, certain sequences of functions can create surprising results when limits and integrals are combined.

For more advanced needs, mathematicians often use the Lebesgue integral instead, which works well even when the Riemann integral runs into problems. The Lebesgue integral agrees with the Riemann integral whenever both are defined. Another generalization is the Henstock–Kurzweil integral, which builds directly on the ideas of the Riemann integral. The Riemann–Stieltjes integral changes how we measure the length of intervals in the integration process.

In multivariable calculus, the ideas of the Riemann integral are used to define multiple integrals for functions with several inputs.

Main articles: Lebesgue integral, Henstock–Kurzweil integral, Riemann–Stieltjes integral, Multiple integral

Comparison with other theories of integration

The Riemann integral works well for many everyday uses, but it has some limits for more advanced math. Other types of integrals, like the Riemann–Stieltjes integral and the Lebesgue integral, can handle more complicated situations. The gauge integral is another option that combines ideas from both Riemann and Lebesgue integrals.

In classrooms, teachers sometimes use the Darboux integral to teach about integrals because it is simpler and gives the same answers as the Riemann integral. Some thinkers believe the gauge integral might be a better starting point for learning about integrals.