Riemann integral

The Riemann integral is a key idea in mathematics, especially in real analysis. It was created by a mathematician named Bernhard Riemann and first shared at the University of Göttingen in 1854.

This integral helps us define the area under a curve and solve many practical problems.

For many useful functions, we can find the value of the Riemann integral using the fundamental theorem of calculus. We can also estimate these integrals with numerical methods or simulate them using Monte Carlo integration. Riemann’s work gives mathematicians and scientists strong tools for studying continuous functions.

Overview

The Riemann integral is a way to find the area under a curve on a graph. It was created by a mathematician named Bernhard Riemann. It helps us add up very small pieces of area to find the total area between two points.

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

Main article: Riemann sum

Similar concepts

The Riemann integral can also be defined as the Darboux integral. This is simpler and works for the same functions. Some books use special ways to calculate integrals, like "left-hand" and "right-hand" sums. These can work if used carefully.

But using both regular intervals and left-hand or right-hand sums can sometimes give wrong answers. The Riemann integral avoids these problems by not integrating certain functions. The Lebesgue integral handles these cases in a different way.

Properties

The Riemann integral has a property called linearity. This means that if you add two functions 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 works well with basic math operations.

Because the integral gives a single number for a function, it is a useful tool for functions that can be integrated this way.

Integrability

A bounded function on an interval is Riemann integrable if it is continuous almost everywhere, meaning it is continuous except 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 used with functions that have values in higher-dimensional spaces, like the space of all coordinates (Rⁿ). This works by integrating each coordinate separately.

The Riemann integral is usually for functions on finite intervals. To use it on very large intervals, we can find the integral as a limit of integrals over bigger and bigger finite intervals. This can be tricky and does not always match other ways of finding integrals.

For more advanced needs, mathematicians often use the Lebesgue integral. The Lebesgue integral works well even when the Riemann integral has problems. The Lebesgue integral agrees with the Riemann integral when both can be used. Another type is the Henstock–Kurzweil integral, which builds on the ideas of the Riemann integral. The Riemann–Stieltjes integral changes how we measure the length of intervals.

In multivariable calculus, the ideas of the Riemann integral help 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 is useful for many everyday math problems. But it has some limits for more advanced math. Other types of integrals can handle more complicated situations. These include the Riemann–Stieltjes integral, the Lebesgue integral, and the gauge integral.

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