- A Riemann sum is a certain kind of approximation of an integral by a finite sum.
- One 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, etc.) 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.
- 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.