It is a huge time saver for people doing applied math and needing quick answers to questions in order to iterate (I do have a soft spot for SymPy if you need to answer several related questions).
Shows up regularly in games of various sorts, both making and breaking. Imagine for some simple contrived example that:
Attacking in melee yields a damage of F(strength), let's say F(x^2)
Attacking in range yields a damage of G(dexterity), let's say F(x*10)
It takes 10 attacks to level up to the next tier. Who will output the most damage on their way from tier 0 to 14? Seems like a simple question, but that number is annoying enough to not be obvious, and an integral gives the answer easily. Integrated [0,14] [x^2 = 915] [10x = 980]. So the linear damage still just outpaces the exponential (in terms of total damage done). Their integrals are equal at 15, after which point the exponential takes the lead.
> While researching this article,[5] I managed to lock up my copy of Mathematica several times on balloon-related differential equations, and subsequently got my IP address banned from Wolfram|Alpha for making too many requests. The ban-appeal form asked me to explain what task I was performing that necessitated so many queries, so this is what I put: ...
> Recently the author of xkcd, Randall Munroe, was asked the question of how long it would be necessary for someone to fall in order to jump out of an airplane, fill a large balloon with helium while falling, and land safely. Randall unfortunately ran into some difficulties with completing his calculation, including getting his IP address banned by Wolfram|Alpha. (No worries: we received his request and have already fixed that.)
I don't know about integrals, but I use it for calculating derivatives, which are useful in Jacobians, which are useful for controlling dynamic systems.