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When we're integrating a non-negative function from *a* to *b*, the integral **can be thought of as the "area under the curve" of the function. However, most of the time we can't count on having a non-negative function to integrate.**

Assume *f* is a function that's allowed to take on negative values, and we're integrating from *a* to *b* with *a *< *b*. Then is the weighted sum of the areas between the graph of *f* and the *x*-axis. We look at all areas between *f* and the *x*-axis. If they're on top of the *x*-axis we count them positively. If they're below the *x*-axis we count them negatively.

In other words, we add all the areas on top of the *x*-axis, then subtract all the areas below the *x*-axis.

### 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*.### General Riemann Sums

For those functions whose integrals we can't find exactly, we can still use the left-hand sum, right-hand sum, midpoint sum and trapezoid sum to estimate their integrals.

The left-hand, right-hand, and midpoint sums are examples of Riemann Sums.

A

**Riemann Sum**is any sum you get when you split up [*a*,*b*] into sub-intervals. The intervals don't necessarily all have to be the same size. Draw a rectangle on the sub-interval using a value of the function on that sub-interval for the rectangle "height."We put "height" in quotes because now the functions are allowed to take on negative values. While we can't have negative heights, we can think of a function value as a weighted height. If the weighted height is negative and the sub-interval goes from left to right, then the weighted area will also be negative.

Whether the LHS, RHS, MID, and TRAP are over- or under-estimates doesn't depend at all on whether

*f*is positive or negative.### Sample Problem

Let

*f*be negative and increasing. The right-hand sum rectangles don't cover enough area, but when*f*is negative this means the right-hand sum will be less negative than the actual integral. Therefore the right-hand sum will be an over-estimate.In order to determine if your sum gives an over-or under-estimate, you can rely on the things we learned earlier:

- If
*f*is increasing then the left-hand sum is an under-estimate and the right-hand sum is an over-estimate.

- If
*f*is decreasing then the left-hand sum is an over-estimate and the right-hand sum is an under-estimate.

- If
*f*is concave up then the trapezoid sum is an over-estimate and the midpoint sum is an under-estimate.

- If
*f*is concave down then the trapezoid sum is an under-estimate and the midpoint sum is an over-estimate.

It doesn't matter for any of these if

*f*is positive or negative. Also, remember that you don't need to remember all of this. You can get by with this list and knowing how to draw graphs.- If

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