This applet computes Riemann sums for a user-defined function and draws a graph of the function as well as a graphical represenatation of the approximations. It can handle exponential, logarithmic, and trigonometric functions, as well as composites of these, and many other functions. The applet is from the author's Java Tools page, accessed from the Interactive Real Analysis page. Because it comes from a real analysis course, it uses terms such as "norm of a partition" and "general sum" that may not be familiar in a calculus class. However, the partitions are all regular -- "general sum" appears to mean that the sample points are chosen randomly in the subintervals.

Bert G. Wachsmuth is in the Department of Mathematics and Computer Science at Seton Hall University.

INTENDED USES:

This applet can be used to illustrate how Riemann sums approximate the value of a definite integral. The user can define the function to be integrated, including piecewise defined functions, and choose any combination of "right", "left", "upper", "lower", or "general" Riemann sum. The applet can be used by instructors for demonstration purposes or by students to explore the relationship between the various Riemann sums and their relation to the value of the integral.

APPROPRIATE COURSES:

This applet and its enclosing project "Interactive Real Analysis" are suitable for analysis courses at the Calculus level and above, including Advanced Calculus and Real Analysis, as well as for self-study. Parts of Interactive Real Analysis are suitable for courses in Set Theory or Foundations of Mathematics.

Open Integrator in a new window

SOFTWARE SPECIFICATIONS: Java Virtual Machine 1.1.8, which is included in Netscape 4.5 (or better) and Internet Explorer 4.0 (or better). All operating systems that support Netscape 4.5 or Internet Explorer 4.0 or better (both with Java enabled) are supported, as well as all operating systems that support a Java Virtual Machine 1.1.8 or better. Operating systems used in testing: Windows 98, MacOS 9.0.3; browsers: Internet Explorer 5, Netscape 4.7.

AUTHOR'S STATEMENT: The applet is part of an elaborate project entitled "Interactive Real Analysis," an online textbook on Real Analysis or Advanced Calculus at the undergraduate level. Interactive Real Analysis covers sets, sequences, series, continuity, differentiability, integrability (Riemann and Lebesgue), topology, and more, and includes many additional applets to illustrate and investigate other topics relevant to Real Analysis. The project is currently in use at many institutions world-wide as a supplement to existing courses or to support self-study efforts. Interactive Real Analysis can be found here.