- Topics At a Glance
- Left-Hand Sum
- Right-Hand Sum
- Comparing Right- and Left-Hand Sums
- Error in Left- and Right-Hand Sums
- Midpoint Sum
- Midpoint Sums with Shortcuts
- Over or Under Estimates
- Trapezoid Sum
- Trapezoid Sum with Shortcuts
- Over or Under Estimates
- Comparison of Sums
- Definite Integrals of Non-Negative Functions
**Definite Integrals of Real-Valued Functions****Conditions for Integration**- General Riemann Sums
- Properties of Definite Integrals
- Single-Function Properties
- Talking About Two Functions
- Thinking Backwards
- Average Value
- Averages with Numbers
- Averages with Functions
- In the Real World
- I Like Abstract Stuff; Why Should I Care?
- How to Solve a Math Problem

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:

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*.