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.
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.