One very practical place I've seen where this is a problem is in monte carlo simulation. A lot of people think of the problem in terms of the monte carlo simul

One very practical place I've seen where this is a problem is in monte carlo simulation. A lot of people think of the problem in terms of the monte carlo simulation. As a result, they forget that the monte carlo simulation is actually an approximation of an integral. If you remember this fact, there's a lot of tricks you can do. You can analytically evaluate parts of the integral. You can replace monte carlo with a more accurate quadrature rule; e.g. Simpson's rule in low dimensions, quasimonte carlo in high. But you'll never do any of that if you don't distinguish between the integration technique (anti-derivative trick, monte carlo, etc) and the integral itself.