It seems that even at Caltech, most students come in not knowing how to use an integral as accumulator (e.g. to calculate a moment of inertia in physics) and the caltech freshman calculus course doesn't help.
To quote my friend Sanjoy Mahajan, physics TA there: Currently, you can just forget about using calculus. Most of the students don't know what an integral is, even if they recognize the symbols. (The Math 1 class is no help, it's the same as 43H,
[43H is Stanford's freshman honors calculus course, aimed at training people in theoretical math, using books with minimal if any connection to applications; in fact, their idea of an "application" is when you actually *evaluate* an integral in, say, three-dimensional Euclidean space, as opposed to talking about integrals in more abstract ways. And now back to the quote...]
where you mostly integrate notation, and forget any math you might have known.) So they don't know how to set up an integral, even if they can "do an integral". When you teach integration, do you use examples like CM, or moment of inertia?
Anyway, I thought you'd like to know that good teachers at Caltech feel the same frustrations with symbol-manipulation-focused calculus as a lot of the good teachers on this list, and that the integral as accumulator idea is one that they want to see emphasized too.