When I first saw integration techniques in calculus, I thought they were a waste of time because software packages could do any integral I could do by hand. Besides, you can always just use Simpson’s rule to compute integrals numerically.

In short, I thought symbolic integration was useless and numerical integration was trivial. Of course **I was wrong on both accounts**. I’ve solved numerous problems at work by being able compute an integral in closed form, and I’ve had a lot of fun cracking challenging numerical integration problems.

Many of the things I thought were impractical when I was in college have turned out to be very practical. And many things I thought would be supremely useful I have yet to apply. Paying too much attention to what is “practical” can be self-defeating. Pragmatism is impractical.

Mathcad will do symbollic integration. So it is still a waste of time to do much of it by hand. Much of the complex problems will be solved in finite element analysis. A lot more focus should be on how to properly model a problem. I find few good examples on how to model complex situations in physics, there are too many shortcuts in my physics book like short angle approximation, infinite lengths, and using circles rather than any other function for a curve, just because it’s easy to do the integral.