# 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.
- 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.

## Essential Questions

Essential Questions

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

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