David Little
Mathematics Department
Penn State University
Eberly College of Science
University Park, PA 16802
Office: 403 McAllister
Phone: (814) 865-3329
Fax: (814) 865-3735
e-mail:dlittle@psu.edu

Methods of Integration: Substitution

The following applet can be used to examine the substitution rule for definite integrals: Simply enter the functions f(g(x))g'(x) and f(u) and the values a, b, c and d. The applet automatically calculates the corresponding areas between each curve and the x-axis. If c=g(a) and d=g(b), then the areas are identical, as predicted by the substitution rule for definite integrals.

The values a, b, c and d can be changed by simply typing a new value, such as "1.2345", "pi/2", "sqrt(5)+cos(3)", etc. You may also change these values by using the up/down arrow keys or dragging the corresponding point left or right. To move the center of the graph, simply drag any point to a new location. To label the x-axis in radians (i.e. multiples of pi), click on the graph and press "control-r". To switch back, simply press "control-r" again.

Here is a list of functions that can be used with this applet.

Note that the areas given by the applet are approximated via Simpson's rule with n=10,000 and are thus subject to the corresponding error bounds.