Wolfram Computation Meets Knowledge

Date Archive: 2008 November

Announcements & Events

Surprise! Mathematica 7.0 Released Today!

In the middle of last year, we finished our decade-long project to reinvent Mathematica, and we released Mathematica 6. We introduced a great many highly visible innovations in Mathematica 6—like dynamic interactivity and computable data. But we were also building a quite unprecedented platform for developing software. And even long before Mathematica 6 was released, we […]

Computation & Analysis

Visualizing Integrals

Calculus II is one of my favorite classes to teach, and the course I’ve probably taught more than any other. One reason for its special place in my heart is that it begins on the first day of class with a straightforward, easily stated, yet mathematically rich question: what is the area of a curved region? Triangles and rectangles—figures with straight sides—have simple area formulas whose derivation is clear. More complicated polygons can be carved up into pieces that are triangles and rectangles. But how does one go about finding the area of a blob?

After simplifying the blob to be a rectangle whose top side has been replaced with a curve, the stage is set for one of the classic constructions in calculus. The area of our simplified blob, reinterpreted as the area under the graph of a function is approximated using a series of rectangles. The approximation is obtained by partitioning the x axis, thus slicing the region into narrow strips, then approximating each strip with a rectangle and summing all the resulting approximations to produce a Riemann sum. Taking a limit of this process by using more and narrower rectangles produces the Riemann integral that forms the centerpiece of Calculus II. Several Demonstrations from the Wolfram Demonstrations Project, including "Riemann Sums" by Ed Pegg Jr, "Common Methods of Estimating the Area under a Curve" by Scott Liao and "Riemann Sums: A Simple Illustration" by Phil Ramsden show that this construction and images like the one below from "Riemann Sums" are part of the iconography of calculus.