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All of these summations are starting to feel like Rube Goldberg Machines. Granted, Rube Goldberg Machines are awesome, but do we seriously need this many methods to sum up intervals? Trust us, they are all useful in their own way. Just one big one to go; call it the grand finale.
A trapezoid sum is different from a left-hand sum, right-hand sum, or midpoint sum. Instead of drawing a rectangle on each sub-interval, we draw a trapezoid on each sub-interval. We do this by connecting the points on the function at the endpoints of the sub-interval.
First, a note on the area of trapezoids.
For a trapezoid that looks like this,
the area of the trapezoid is the average of the areas of two rectangles.
Thanks to the distributive property, this can be rewritten as
Now we'll explore sums using trapezoids through the examples.
The trapezoid sum is a good one to have some shortcuts for. We'll call the trapezoid sum with n sub-intervals TRAP(n).
Here's our favorite shortcut: TRAP(n) is the average of LHS(n) and RHS(n).
These rectangles are, respectively a left-hand sum and a right-hand sum!
Remember that the trapezoid sum is the average of the left- and right-hand sums. However, there's an even shorter way to get a trapezoid sum out of your calculator.
LHS(n) = [f(x0) + f(x1) + ... + f(xn – 1)]Δx
RHS(n) = [f(x1) + ... + f(xn – 1) + f(xn)]Δx.
The trapezoid sum is the average of the right- and left-hand sums, so
This is kind of a mess. It gets better if we factor out the Δx:
Now look carefully at what we have inside the parentheses. The quantities f (x0) and f (xn) only show up once each, because f (x0) is only used in the left-hand sum and
f (xn) is only used in the right-hand sum:
However, every term from f(x1) to f(xn – 1) is used in both the left-hand sum and right-hand sum, so each of these terms will show up twice each.
If we're estimating the area between f and the x-axis on [a, b] with TRAP(n) the first thing we do is divide [a, b] up into n equal sub-intervals and find the endpoints.
The value f(x0) is only used as a height of the left-most trapezoid. Similarly, the value f(xn) is only used as a height of the right-most trapezoid. However, the value of f at every endpoint in between these shows up in two trapezoids.
When we add the areas of all these trapezoids we get
Factoring out the and the Δx gives us
Now we have a much better way to find a trapezoid sum:
Like the midpoint sum, whether the trapezoid sum gives an under- or over-estimate depends on the concavity of the function.
If our function is concave up, the trapezoids we get from the trapezoid rule cover too much area, so we get an overestimate.
If our function is concave down, the trapezoids don't cover enough area, so we get an underestimate.
If our function has no concavity (is a straight line), the trapezoid sum is just right. Which porridge will you pick, Goldilocks?
Whether the function is increasing or decreasing doesn't matter.