Riemann Sums

  • 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.

Essential Questions

How can the area of a plane region be estimated and found precisely?

Prerequisite Understanding