Say you have a calculus classroom full of liberal-arts majors who are not particularly mathematically inclined. Your goal is NOT to teach them everything on a list of topics that will be needed in other courses (either later calculus courses or courses in physics or engineering or statistics or biology or other subjects to which calculus is applied). (As anyone with any common sense would do) you will include only ten percent or less of the topics in the usual lists and perhaps examine each included topic in more depth than it might normally get, but perhaps also proceed more slowly than you would if you needed to complete all the topics in the usual list. Rather, your goal is to impress them with (1) the ways in which calculus has played a role and continues to play a role in the world --- in the sciences and engineering and philosophy, etc. --- and with (2) the fact that calculus is a considerable intellectual and aesthetic achievement. This might necessiate presenting a few applications in the sciences, not usually found in the first-year calculus text, rather than concentrating on techniques.
My question is: What topics would you include in such an (in the present day) unusual course?
I would include the formulation of the Riemann integral, and include many applications such as calculating area, volume, center of mass, work, etc.. Approximating and taking a limit is, in my opinion, a beautiful idea that also turns out to be very fruitful.