Definition of Definite Integrals - Riemann Sums
This is an applet to explore the definition of the definite integral.
Function f(x) has the form
f(x) = a*x2+d ,
where a and d can be changed in order to experiment with several functions. The experiment is started by approximating the area under the curve from x = 1 to x = -1 using 5 rectangles whose heights are equal to the value of the function at points within these rectangles. Then the mesh size h of the rectangles is decreased. The sampling points are selected randomly within an interval. When you first start the applet a = -2 and d = 2 and there are 5 rectangles.
1 - click on the button above "click here to start" and MAXIMIZE the window obtained.
2 - The approximation of the area starts with 5 rectangles from x=-1 to x=1. The horizontal dimension h of the rectangles can be made smaller by clicking on the button "Decrease h". As you click on this button note the convergence of the area (bottom,left) to a finite value and the geometrical behavior of the area under the rectangles. The Reset button is used when a new experiment is to be started.
3 - Change a and d and note the geometrical behavior of the area under the rectangles and also the convergence of the area. Compare the area as approximated by this applet with the exact value using definite integrals.