# The definite integral as the limit of a Riemann sum

- Author:
- Luca Moroni

- Topic:
- Area, Definite Integral, Trapezoid

This worksheet examines the constructions and accuracies of different integral approximation methods and its relations with the exact integration provided by the primitive function.
The check-boxes on the left side allow you to view the exact area and to toggle between rectangles with right, left and middle points and with the trapezoidal approximations.
The extremes of integration of the function where .
It can be shown that, when , the area equals the definite integral
since it is
The "Show primitive function" check-box allows to see this construction.
There are two alternative functions

*a*and*b*and the number of intervals*n*can be set with the sliders in the upper left side. The actual area and that of each approximation are shown at the bottom left side of the worksheet. The actual area can be exactly calculated with the primitive*f*_{1}(*x*) and*f*_{2}(*x*) already set in the workbook. With the check-box "*Alternative function f*" it's possible to switch from the first one to the second one._{2}(x)**Credits**:

*Christopher Stover*. This worksheet builds up from his original worksheet "Rectangular and Trapezoidal Integral Approximations", once available on http://personal.bgsu.edu/~stoverc/Geogebra/index.html