November 18, 2008 — Stephen Wolfram
And even long before Mathematica 6 was released, we were already working on versions of Mathematica well beyond 6.
And something remarkable was happening. There’d been all sorts of areas we’d talked about someday being in Mathematica. But they’d always seemed far off.
Well, now, suddenly, lots of them seemed like they were within reach. It seemed as if everything we’d built into Mathematica was coming together to make a huge number of new things possible.
All over our company, efforts were starting up to build remarkable things.
It was crucial that over the years, we’d invested a huge amount in creating long-term systems for organizing our software development efforts. So we were able to take those remarkable things that were being built, and flow them into Mathematica.
And at some point, we realized we just couldn’t wait any longer. Even though Mathematica 6 had come out only last year, we had assembled so much new functionality that we just had to release Mathematica 7.
November 13, 2008 — Chris Boucher, Consultant, Special Projects Group
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.