three differentiaion / integration relationships I come across

displacement - speed - acceleration - rate of change of acceleration

displacement - distributed load - shear force

capacitors integrate current, this can be used in sampling.

You use it to make money, because the gatekeepers of social mobility in western society have decided (wisely or otherwise) that demonstrating an understanding of calculus is a prerequisite to being qualified in technical fields, many of which rarely require you to actually use calculus.

While it doesn't involve solving equations on paper, a lot of times when I'm driving I tend to muse on how applying my foot on the pedal influences my acceleration, how that translates into velocity, and how that in turn determines my distance traveled. I use this to restrain myself from the strong temptation to drive aggressively and weave in and out of traffic since, unless I'm driving long distances, the difference in arrival time is negligible.