Riemann sum

A Riemann sum is a mathematical method used to approximate an integral. It works by using a finite sum to estimate a value that might be difficult to calculate exactly. This process is a cornerstone of numerical integration, which is the practice of approximating the area under functions or lines on a graph. Because of this, the method is often called the rectangle rule. Beyond just measuring area, Riemann sums can also be used to approximate the length of various curves.

The mechanism of a Riemann sum relies on partitioning a region into smaller shapes. These shapes can be rectangles, trapezoids, parabolas, or even cubics. To perform the calculation, you first divide a closed interval of real numbers into several subintervals. You then calculate the area for each individual shape within those subintervals. Finally, you add all of these small areas together to reach a total sum. Because these small shapes usually do not match the original region's shape perfectly, the sum will differ from the true area. This difference is known as an error. You can reduce this error by dividing the region more finely into smaller and smaller shapes. As the width of these shapes approaches zero, the sum approaches the Riemann integral.

There are several distinct types of Riemann sums depending on how you choose the height of your shapes. If you use the value of the function at the left endpoint of every subinterval, you have a left Riemann sum. Using the right endpoint for every subinterval results in a right Riemann sum. The midpoint rule uses the value at the center of each subinterval. This middle Riemann sum is often the most accurate approach among these basic methods. Another option is the upper rule, or upper Darboux sum, which uses the supremum of the function over each subinterval. Conversely, the lower rule, or lower Darboux sum, uses the infimum. Any Riemann sum on a specific partition will always fall between these upper and lower Darboux sums.

Historically, this method is named after Bernhard Riemann. He was a nineteenth-century German mathematician. His work provided the foundation for understanding how these sums relate to the definite Riemann integral. A function is considered Riemann integrable if all possible Riemann sums converge to the same value as the partition becomes finer. This means that as the maximum size of the subintervals shrinks toward zero, the different methods will eventually yield the same result.

Specific mathematical rules offer different levels of precision and error. The left and right rules are simple but can be prone to overestimation or underestimation. For example, if a function is monotonically increasing, a right Riemann sum will overestimate the area. The error for these basic rules is related to the maximum absolute value of the function's derivative. The trapezoidal rule provides a different approach by taking the average of the left and right Riemann sums. This method uses the area formula for a trapezium with parallel sides. While the trapezoidal rule is more advanced than basic rectangle rules, it is still followed in complexity by Simpson's rule and Newton-Cotes formulas.

Riemann sums are not limited to one-dimensional lines; they can be generalized to higher dimensions. In two dimensions, the domain is divided into many two-dimensional cells. Each cell has an area, and the sum is calculated by multiplying the function value by the area of each cell. In three dimensions, the process uses three-dimensional cells to measure volume. This pattern continues for any number of dimensions, known as an n-dimensional Riemann sum. In these cases, the process involves a "measure," which is a function that gives a size to a set. In one dimension, this measure is length, in two dimensions it is area, and in three dimensions it is volume.

The significance of this method is most apparent when a function does not have an easy-to-find anti-derivative. In such cases, finding a closed-form solution through the fundamental theorem of calculus is difficult or impossible. Riemann sums provide a practical, numerical way to solve these complex problems. By breaking a complex, continuous curve into manageable, discrete pieces, mathematicians can estimate values that would otherwise remain unknown. This makes the Riemann sum a vital tool in both theoretical mathematics and practical numerical analysis.