Midpoint Rule: Formula & Example

Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

Finding an estimate for the area under a curve is a task well-suited to the midpoint rule. In this lesson we use an example to show the general idea of this formula and how to use it.

The Midpoint Rule: A Lawn Seeding Analogy

You are headed to the store to buy some lawn seed. To determine the amount of seed to buy, we need the area of the lawn. What if our lawn has an unusual shape? Finding the area may not be obvious! It turns out that we can get a good estimate of the area by using rectangles and something called the midpoint rule. Through the use of an example, we will show how this is done.

Getting the Idea

Let's say we would like to find the area under the following curve. To be more precise, let's say we want the area under the curve from x = 2 to x = 10. The desired area is the shaded region:


To explain the overall idea, we are going to start with a small number of rectangles. Let's say we use four rectangles. We position the rectangles so that they do not overlap but cover the desired area. The top portion of the middle of each rectangle just touches the curve. Also, we have decided to have the width of each rectangle be the same. Here's what all of that looks like:


We know how to find the area of rectangles. The area of a rectangle is simply the length times the width.

Let's look carefully at the first of these rectangles.


The endpoints of the first rectangle are at x = 2 and x = 4. Thus, the point in the middle is at x = 3. We will use a subscript label with x to represent the midpoint for rectangle 1. The midpoint for the first rectangle is labeled x1 and is equal to 3. The curve is defined by some function f(x). Thus, f(x1) is the function f(x) evaluated at x1. Note that positioning the rectangle with midpoint on the curve is a good thing. The area missed on the left balances with the extra area gained on the right. Nice!

We see that the length of the rectangle is f(x1) and the width is Δx. With our lawn, we could use a tape measure and determine these distances. Or we could use the graph. We see that the curve for f(x) crosses the midpoint at 1.5. We will use 1.5 as the length for this first rectangle.

The area of the first rectangle is the product of f(x1) times Δx. That is, the product of 1.5 and 2 which gives us an area of 3.

By adding together the areas of each of the four rectangles, we can get a good approximation to the entire area under the curve. Intuitively, the more rectangles we use, the better will be our approximation. For four rectangles, the entire area is 9.4. But 9.4 what? Each unit on the graph represents 10 feet on our actual lawn. One unit times one unit represents 10x10 = 100 square feet. Thus, our estimate of 9.4 on the graph is actually 940 square feet on the lawn. That's a one pound bag of grass seed!

Getting More Mathematical

We will continue with our example with four rectangles and then generalize to any number of rectangles. We found that the approximate area A is the sum of the areas of each of the four rectangles. This could be written as


where x1, x2, x3 and x4 are the midpoints for each rectangle. In our example, these midpoints are 3, 5, 7 and 9. The corresponding rectangle lengths are 1.5, 1, .8 and 1.4. Remember that if we know the function f(x), then these lengths are the function evaluated at each midpoint. Let's do one of these in detail. The actual function for this curve is


Evaluated at the first midpoint, we let x = 3. This gives


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