To approximate the definite integral we split the region up into many thin slices. For each slice we may approximate its area using rectandles, in which case we could either use right-endpoints, left-endpoints, or midpoints to work out the height of the approximating rectangle (seen in blue). We could also use trapezoids to approximate each slice (seen in red). Drag the slider labeled n to increase the number of slices used.
On the other hand, we may group two sub-intervals together and use a parabola to approximate the top of the slice. This is a technique known as Simpson's Method (or Simpson's Rule).
Lastly, we could use a Maclaurin Series Expansion of the function (use the slider to set the order, or number of terms) to approximate the integral
If you wish to change the function f, say to sin(x), then just type f(x)=sin(x) in the input field at the bottom of the applet.
This applet is a modification of "Approximate Integration" by J Mulholland (http://www.geogebratube.org/student/m14086)