The definite integral as the limit of a Riemann sum

The definite integral

This worksheet, made with Geogebra, examines the constructions and accuracies of different approximation methods (Riemann sums) for the area of a region underneath a curve and their deviations from the exact value provided by the definite integral:

    \[\int\limits_a^b {f\left( x \right)} dx  = F\left( b \right) - F\left( a \right)\]

where F\left( x \right) is a primitive of the function f\left( x \right).

Use of the applet

The check-boxes on the left side allow you to view the exact area and to toggle between rectangular approximations with right, left and middle points and with the trapezoidal approximations.

The extremes of integration 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 area can be exactly calculated with the primitive Y=F(x) of the function f(x) where F'\left(x \right) =f\left( x \right).

In fact, using for the function Y = F\left( x \right) the definition of the differential \Delta Y = F'\left( x \right)\Delta x, it’s possible to prove that

    \[ \sum{f\left( x \right) }\Delta x=\sum{F'\left( x \right) }\Delta x=\sum{\Delta Y =F\left(b \right)-F\left(a \right) }\]

And so, when \Delta x \to 0,

    \[ \sum{f\left( x \right) }\Delta x \to \int_{a}^{b}{f\left( x \right)dx }=F\left(b \right) -F\left( a \right)\]

The “Show primitive function” check-box allows to see this construction and to verify that the vlaue of the area is given by the difference F\left(b \right)-F\left(a \right).
The primitive function F\left( x \right) is defined up to an additive constant C (whose value can be changed with the slider that is shown when the “Show primitive function” check-box is checked).
Anyway the value of F\left(b \right)-F\left(a \right), that is the value of the area, doesn’t change whit changes of the constant C.

There are two alternative functions f_1(x) and f_2(x) already set in the workbook.
With the check-box Alternative function f_2(x) it’s possible to switch from the first one to the second one.
The main function f_1(x) can be changed in the input bar by inserting its definition (i.e. f(x) =x^3-cos(x)).

The button in the top right corner can be used to reset the construction to its default values.

Credits: Christopher Stover. This worksheet builds up from his original worksheet “Rectangular and Trapezoidal Integral Approximations” (,