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 approximating the area of functions or lines on a graph, but also the length of curves and other approximations. The sum is calculated by partitioning the region into shapes (rectangles, trapezoids, parabolas, or cubics) 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 be a function defined on a closed interval of the real numbers, , and as a partition of , that is A Riemann sum of over with partition is defined as where and . One might produce different Riemann sums depending on which 's are chosen. In the end this will not matter, if the function is Riemann integrable, when the difference or width of the summands approaches zero. Specific choices of give different types of Riemann sums: If for all i, the method is the left rule and gives a left Riemann sum. If for all i, the method is the right rule and gives a right Riemann sum. If for all i, the method is the midpoint rule and gives a middle Riemann sum. If (that is, the supremum of over ), the method is the upper rule and gives an upper Riemann sum or upper Darboux sum. If (that is, the infimum of f over ), the method is the lower rule and gives a lower Riemann sum or lower Darboux sum.