Riemann Sums for Functions of Two Variables

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Visualize and calculate a Riemann sum for a real–valued function of two real variables. The x and y axes are partitioned into subintervals of equal width. Set the point on each subrectangle where the function is evaluated to determine the height of the rectangular solid constructed over that subrectangle.

Contributed by: Bruce Torrence (March 2011)
Open content licensed under CC BY-NC-SA


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This Demonstration is meant for students of multivariable calculus. It illustrates the concept of a Riemann sum for a real–valued function of two real variables. Choose one of four functions. Then you can freely select a uniform partition of either the x or y axis into 5, 10, 20, or 40 parts. You may also select the point on each subrectangle at which the function is evaluated (to determine the height of the rectangular solid to be constructed over that subrectangle). Initially, the function is evaluated at the center. This feature is included to show that as the partition becomes finer, the choice of this point becomes less consequential, suggesting that in the limit it simply does not matter.



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