From 11:00PM PDT on Friday, July 1 until 5:00AM PDT on Saturday, July 2, the Shmoop engineering elves will be making tweaks and improvements to the site. That means Shmoop will be unavailable for use during that time. Thanks for your patience!

# Conditions for Integration

We can only integrate real-valued functions that are reasonably well-behaved. No Dance Moms allowed. If we want to take the integral of f(x) on [a, b], there can't be any point in [a,b] where f zooms off to infinity. When it comes to definite integrals, this is bad:

Depending on how the function behaves near the asymptote, we may still be able to take the integral, but it won't be a definite integral. These are called improper integrals, but we won't go in depth with those guys just yet.

Having a continuous function is great, but if it's only discontinuous at a few points, that's allowed too. For example, what if f(x) = 5 for all x ≠ 1 but is undefined at x = 1? That function looks like this:

If we want to integrate f from 0 to 2 there's one little spot where f isn't defined. That means the integral needs to account for all the area in this rectangle except for the line at x = 2:

Since a line doesn't have any area, taking out that line doesn't take away any area from the rectangle. This means it's not a problem for f to be undefined at that one point.

Similarly, it's not a problem for f to be undefined at ten separate points. Each individual line has no area, so the ten lines together have no area. It's also not a problem for f to be undefined at 100 separate points. Or a million.

When we can find the integral of a function on [a, b], we say that function is integrable on [a, b]. If a function is integrable for any interval we pick we say that function is integrable.