You can approximate the area under a curve using rectangles. To do
this, divide the base interval into pieces (*
subintervals*). Then on each subinterval, build a rectangle that
goes up to the curve.

What does it mean to "go up to the curve"? You have to make a choice about how the height of each rectangle depends on the curve. In the picture above, for my rectangle height I always used the height of the curve above the left-hand endpoint of each subinterval.

In the picture above, I used subintervals of different sizes. For simplicity, you will often use subintervals of the same size --- so your rectangles all have the same width. In the picture below, I've used 8 rectangles of equal widths, and for my rectangle height I always used the height of the curve above the right-hand endpoint of each subinterval.

Here's an example with a specific function. I'll use . In each case, I'll use the base interval divided into 6 equal subintervals:

Here is the picture if I use the left-hand endpoint of each subinterval to get the height of each rectangle:

The sum of the areas of the rectangles is:

Notice that the rectangle width 0.25 factors out of the sum --- you add up the f's, then multiply by 0.25. This will always be possible if you use subintervals of equal length.

Here is the picture if I use the right-hand endpoint of each subinterval to get the height of each rectangle:

In this case, the sum of the areas of the rectangles is

Here is the picture if I use the midpoint of each subinterval to get the height of each rectangle:

The midpoints of the subintervals are:

In this case, the sum of the areas of the rectangles is

By comparison, the actual area under the curve is around 0.778238.

You can get better approximations by taking more rectangles. For example, here is the left-hand endpoint picture with 50 rectangles:

Notice how much better the rectangles approximate the area under the curve.

With 200 rectangles, the left-hand endpoint sum is 0.775311, the right-hand endpoint sum is 0.781147, and the midpoint sum is 0.778242. The three values are close to the actual value 0.778238.

* Example.* Approximate the area under for , using 20
*circumscribed* rectangles of equal width.

*Circumscribed* means that you should use the *largest*
function value on each interval to get the height of a rectangle. My
subintervals are

In general, it can be difficult to determine where the largest function value is. However, by graphing on , you can see that the largest function value for each subinterval occurs at the left-hand endpoint.

So I use

I continue in this fashion, all the way up to

I can write these points as

So my function values are

These are the rectangles heights. Each height is multiplied by a width of 0.1. The total is

Now the sum is in a form you can evaluate on your calculator. You should get 5.53; the actual value is .

* Example.* Approximate the area under from to using 20 equal subintervals and evaluating the
function at the left-hand endpoints.

You can use a calculator to approximate this sum; it's around 0.84799.

In the preceding examples, I've assumed that the subintervals (which give the widths of the rectangle) are the same size. I've also chosen the evaluation points systematically --- left-hand endpoints, right-hand endpoints, midpoints. These are conveniences to make setting up the computations simple.

In general, the subintervals don't have to be the same size, and I don't have to choose the evaluation points systematically. For example, here is an approximation to the area under from to .

This gives an approximate area of 51.84. The actual area is 77.

Copyright 2018 by Bruce Ikenaga