Riemann sum
（重定向自Upper sum）
![Four of the Riemann summation methods for approximating the area under curves. Right and left methods make the approximation using the right and left endpoints of each subinterval, respectively. Maximum and minimum methods make the approximation using the largest and smallest endpoint values of each subinterval, respectively. The values of the sums converge as the subintervals halve from top-left to bottom-right.](/Images/godic/202502/19/Riemann_sum_convergence2333.png")
![Riemann sum methods of x3 over [0,2] using 4 subdivisionsLeftRightMiddleTrapezoidalSimpson's Method](/Images/godic/202502/19/LeftRiemann2.svg2333.png")
![](/Images/godic/202502/19/RightRiemann2.svg2333.png")
![](/Images/godic/202502/19/MidRiemann2.svg2333.png")
In mathematics, a Riemann sum is an approximation that takes the form ![\sum f(x) \Delta x](/Images/godic/202502/19/e348a406d8a39f1c79eb8203d1e0741c2333.png")
. It is named after 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.