In the Real World
Remember all that stuff about left-hand sums and right-hand sums? That's why your calculator works. Your calculator doesn't know what sin(3) is any better than you do, but it's able to take accurate approximations quickly.
We talked about how if you take enough rectangles, you can guarantee that the approximation is within a certain amount of the real answer. A calculator screen only displays ten or so decimal places, so the calculator needs to guarantee that its approximation is correct for as many decimal places as the screen will show.
It's cool to think that you can say your approximation is no more than a certain distance from the real answer even though you don't know what the real answer is.