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 approximating the area of functions or lines on a graph, but also the length of curves and other approximations. Left sumRight sumMiddle 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 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 filled 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 f : [ a , b ] → R {displaystyle f: ightarrow mathbb {R} } be a function defined on a closed interval [ a , b ] {displaystyle } of the real numbers, R {displaystyle mathbb {R} } , and be a partition of I, where A Riemann sum S {displaystyle S} of f over I with partition P is defined as where Δ x i = x i − x i − 1 {displaystyle Delta x_{i}=x_{i}-x_{i-1}} and an x i ∗ ∈ [ x i − 1 , x i ] {displaystyle x_{i}^{*}in } .Notice the use of 'an' instead of 'the' in the previous sentence. Another way of thinking about this asterisk is that you are choosing some random point in this slice, and it does not matter which one; as the difference or width of the slices approaches zero, the difference between any two points in our rectangle slice approaches zero as well. This is due to the fact that the choice of x i ∗ {displaystyle x_{i}^{*}} in the interval [ x i − 1 , x i ] {displaystyle } is arbitrary, so for any given function f defined on an interval I and a fixed partition P, one might produce different Riemann sums depending on which x i ∗ {displaystyle x_{i}^{*}} is chosen, as long as x i − 1 ≤ x i ∗ ≤ x i {displaystyle x_{i-1}leq x_{i}^{*}leq x_{i}} holds true. Specific choices of x i ∗ {displaystyle x_{i}^{*}} give us different types of Riemann sums: All these 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'.