Trapezoid Sum Examples

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?

First we divide the interval [0, 4] into two sub-intervals, as usual. On [0 ,2] we find the value of f at each endpoint of the sub-interval and plot the appropriate points.

The area of this trapezoid is

On [2, 4] we find the value of f at each endpoint of the sub-interval and plot the appropriate points.

The two heights of the trapezoid are f(2) = 5 and f(4) = 17, so the area of this trapezoid is

We add the areas of the trapezoids and estimate that the area of R is

6 + 22 = 28.

Since the trapezoids cover R and then some, this is an overestimate. Graphed, the trapezoids look like this:

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?

We divide [0, 4] into four sub-intervals of width 1. The first trapezoid has heights f(0) = 1 and f(1) = 2.

It's area is

The second trapezoid has heights of 2 (we know this from the previous trapezoid) and f (2) = 5.

It has area

The third trapezoid has heights 5 (known from previous trapezoid) and f (3) = 10.

It has area

.

The third trapezoid has heights 10 (known from previous trapezoid) and f(4) = 17.

It has area

Adding up the areas of the trapezoids, we estimate the area of R to be

1.5 + 3.5 + 7.5 + 13.5 = 26.

This is an overestimate, since again the trapezoids cover all of R and then some.