The 1 is implied

Due to my misspent youth (aka, college the first time around), I am currently taking Calculus III. Now, the last time I took calculus at all is about 2 decades ago, so I have been treating this class like the job it really is. The course started out with vectors and moved to partial derivatives and their applications. This is all fine, but now I am in the midst of double and triple integrals and my brain… well, let’s just say that Swiss cheese is not as holey as my brain feels right about now.

Derivatives are relatively easy and intuitive. They mostly use symbols and letters that through algebra and trigonometry we’ve come to understand. But integration…

double and triple integrals
The attack of the integrals

It feels like a completely different language. Now, one of my classmates had a good question the other day. What are we integrating? Most of the time you have an equation there, so it looks like x2dx. Well, when there is nothing there, the answer is easy but not as obvious. We are integrating the constant: 1. So, the first integral is easy – just plug in the upper bound and subtract out the lower bound.

What makes this interesting to me is it is an excellent example of an instructor being too close to the material sometimes to realize that something isn’t obvious. Over 2 decades ago, I was in a precalculus class in high school where every single one of us failed the first test in differentiation. Why? All of us had passed all of the homework assignments. We had learned how to differentiate without understanding why we were doing it.

It was the power rule. Take the exponent, put it in front of the variable and then subtract 1 from the power.

nxn-1

And that was what we did. All of our homework questions and classroom examples had been the simplest case: x5 The test, however, was full of things like 5x5, and we had missed that simple logical step. All of our previous examples involved multiplying n whether it be 1 or 2 or 5 or another variable with the number 1 and we just hadn’t gotten it.

The implied 1 is true for just about everything. I spend my days helping people set up databases or configure their software, and I have to remind myself what the steps are – not because I have forgotten them, but because I know them so well and they feel so obvious that it is easy to skip over them when I am trying to describe to someone how to do a process.