Riemann sum

In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum. It is named after nineteenth century German mathematician Bernhard Riemann. One very common application is in numerical integration, i.e., approximating the area of functions or lines on a graph, where it is also known as the rectangle rule. It can also be applied for approximating the length of curves and other approximations.

The sum is calculated by partitioning the region into shapes (rectangles, trapezoids, parabolas, or cubics—sometimes infinitesimally small) that together form a region that is similar to the region being measured, then calculating the area for each of these shapes, and finally adding all of these small areas together. This approach can be used to find a numerical approximation for a definite integral even if the fundamental theorem of calculus does not make it easy to find a closed-form solution.

Because the region by the small shapes is usually not exactly the same shape as the region being measured, the Riemann sum will differ from the area being measured. This error can be reduced by dividing up the region more finely, using smaller and smaller shapes. As the shapes get smaller and smaller, the sum approaches the Riemann integral.

Let ${\displaystyle f:[a,b]\to \mathbb {R} }$ be a function defined on a closed interval ${\displaystyle [a,b]}$ of the real numbers, ${\displaystyle \mathbb {R} }$, and ${\displaystyle P=(x_{0},x_{1},\ldots ,x_{n})}$ as a partition of ${\displaystyle [a,b]}$, that is $${\displaystyle a=x_{0}<x_{1}<x_{2}<\dots <x_{n}=b.}$$ A Riemann sum ${\displaystyle S}$ of ${\displaystyle f}$ over ${\displaystyle [a,b]}$ with partition ${\displaystyle P}$ is defined as $${\displaystyle S=\sum _{i=1}^{n}f(x_{i}^{*})\,\Delta x_{i},}$$ where ${\displaystyle \Delta x_{i}=x_{i}-x_{i-1}}$ and ${\displaystyle x_{i}^{*}\in [x_{i-1},x_{i}]}$. One might produce different Riemann sums depending on which ${\displaystyle x_{i}^{*}}$'s are chosen. In the end this will not matter, if the function is Riemann integrable, when the difference or width of the summands ${\displaystyle \Delta x_{i}}$ approaches zero.

Specific choices of ${\displaystyle x_{i}^{*}}$ give different types of Riemann sums:

All these Riemann summation methods are among the most basic ways to accomplish numerical integration. Loosely speaking, a function is Riemann integrable if all Riemann sums converge as the partition "gets finer and finer".

While not derived as a Riemann sum, taking the average of the left and right Riemann sums is the trapezoidal rule and gives a trapezoidal sum. It is one of the simplest of a very general way of approximating integrals using weighted averages. This is followed in complexity by Simpson's rule and Newton–Cotes formulas.

Any Riemann sum on a given partition (that is, for any choice of ${\displaystyle x_{i}^{*}}$ between ${\displaystyle x_{i-1}}$ and ${\displaystyle x_{i}}$) is contained between the lower and upper Darboux sums. This forms the basis of the Darboux integral, which is ultimately equivalent to the Riemann integral.