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.
Example 1
Let f (x) = x2 + 1 and let R be the region between the graph of f and the x-axis on [0, 4]. Use a trapezoid sum with 2 sub-intervals to estimate the area of R. Is this an over-estimate or an under-estimate? |
Example 2
Let f(x) = x2 + 1 and let R be the region between the graph of f and the x-axis on [0, 4]. Use a trapezoid sum with 4 sub-intervals to estimate the area of R. Is this an over-estimate or an under-estimate? |
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Exercise 1
Let 
. Use a trapezoid sum with 3 sub-intervals to estimate the area between f and the x-axis on [0, 9]. Is this an overestimate or underestimate?
Exercise 2
The table below shows partial values of the function f(x). Use a trapezoid sum with 4 sub-intervals to estimate the area between the graph of f and the x-axis on [0, 1].
