Most calculus students might think that if one could compute indefinite integrals, it would always be easy to compute definite ones. After all, they might think, the fundamental theorem of calculus says that one just has to subtract the values of the indefinite integral at the end points to get the definite integral.

So how come inside Mathematica there are thousands of pages of code devoted to working out definite integrals---beyond just subtracting indefinite ones? The answer, as is often the case, is that in the real world of mathematical computation, things are more complicated than one learns in basic mathematics courses. And to get the correct answer one needs to be considerably more sophisticated. In a simple case, subtracting indefinite integrals works just fine. Consider computing the area under a sine curve, which equals

I work on computational algorithms for Mathematica, and I always like to see that what I do is helpful in solving real-world problems. When I heard about the I-35W bridge collapse, I wanted to see if anything could be learned from computing the mechanics of the bridge with Mathematica. Large packages have been written for doing structural computations with Mathematica. But I wanted to start from first principles to try to understand the whole picture. A truss bridge can be thought of as a graph, with trusses as edges and joints as nodes, as in the picture here: