Today in second semester calculus, I found myself stumbling a bit to provide a natural-sounding explanation for all the letters involved in the expression $$ \lim_{t \rightarrow \infty} \int_1^t \frac{1}{x^p} dx, $$ where $p$ is an arbitrary real number. There are some subtle things going on.
- The value of $p$ is fixed throughout the evaluation, but is arbitrarily chosen at the beginning.
- The value of $t$ is fixed as regards the definite integral, but is later pushed toward infinity when the limit is evaluated.
- The value of $x$ is not like either of $p$ or $t$, but rather is the variable of integration. It ranges over the domain $[1,t]$, which again is finite for any fixed $t$.
I think these distinctions arise frequently in analysis, so I'd be interested in advice on making them more concrete to students.
One thing I think might help is to separate the limit and the integral. That is, I would first ask them to consider $\int_1^t \frac{1}{x^p} dx$ for arbitrary (but fixed) $p$ and $t > 1$. In this case, the only thing varying is the variable of integration. Now we have a function $A(t)$ that gives the area under $\frac{1}{x^p}$ on $[1,t]$, and we can ask how this function behaves as $t$ tends to infinity. While this viewpoint is helpful to me, I wonder if it is still too abstract for a typical Calculus student.
