5.1: Approximating Areas
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- Use sigma (summation) notation to calculate sums and powers of integers.
- Use the sum of rectangular areas to approximate the area under a curve.
- Use Riemann sums to approximate area.
Archimedes was fascinated with calculating the areas of various shapes—in other words, the amount of space enclosed by the shape. He used a process that has come to be known as the method of exhaustion, which used smaller and smaller shapes, the areas of which could be calculated exactly, to fill an irregular region and thereby obtain closer and closer approximations to the total area. In this process, an area bounded by curves is filled with rectangles, triangles, and shapes with exact area formulas. These areas are then summed to approximate the area of the curved region.
In this section, we develop techniques to approximate the area between a curve, defined by a function
Let’s start by introducing some notation to make the calculations easier. We then consider the case when
Sigma (Summation) Notation
As mentioned, we will use shapes of known area to approximate the area of an irregular region bounded by curves. This process often requires adding up long strings of numbers. To make it easier to write down these lengthy sums, we look at some new notation here, called sigma notation (also known as summation notation). The Greek capital letter
We could probably skip writing a couple of terms and write
which is better, but still cumbersome. With sigma notation, we write this sum as
which is much more compact. Typically, sigma notation is presented in the form
where
Let’s try a couple of examples of using sigma notation.
- Write in sigma notation and evaluate the sum of terms
for - Write the sum in sigma notation:
Solution
- Write
- The denominator of each term is a perfect square. Using sigma notation, this sum can be written as
.
Write in sigma notation and evaluate the sum of terms
- Hint
-
Use the solving steps in Example
as a guide.
- Answer
-
The properties associated with the summation process are given in the following rule.
Let
We prove properties (ii.) and (iii.) here, and leave proof of the other properties to the Exercises.
(ii.) We have
(iii.) We have
□
A few more formulas for frequently found functions simplify the summation process further. These are shown in the next rule, for sums and powers of integers, and we use them in the next set of examples.
1. The sum of
2. The sum of consecutive integers squared is given by
3. The sum of consecutive integers cubed is given by
Write using sigma notation and evaluate:
- The sum of the terms
for - The sum of the terms
for
Solution
a. Multiplying out
b. Use sigma notation property iv. and the rules for the sum of squared terms and the sum of cubed terms.
Find the sum of the values of
- Hint
-
Use the properties of sigma notation to solve the problem.
- Answer
-
Find the sum of the values of
Solution
Using Equation
Evaluate the sum indicated by the notation
- Hint
-
Use the rule on sum and powers of integers (Equations
- ).
- Answer
-
Approximating Area
Now that we have the necessary notation, we return to the problem at hand: approximating the area under a curve. Let

How do we approximate the area under this curve? The approach is a geometric one. By dividing a region into many small shapes that have known area formulas, we can sum these areas and obtain a reasonable estimate of the true area. We begin by dividing the interval
for
We denote the width of each subinterval with the notation
for
A set of points
We can use this regular partition as the basis of a method for estimating the area under the curve. We next examine two methods: the left-endpoint approximation and the right-endpoint approximation.
On each subinterval

The second method for approximating area under a curve is the right-endpoint approximation. It is almost the same as the left-endpoint approximation, but now the heights of the rectangles are determined by the function values at the right of each subinterval.
Construct a rectangle on each subinterval
The notation

The graphs in Figure
In Figure

In Figure
Use both left-endpoint and right-endpoint approximations to approximate the area under the curve of
Solution
First, divide the interval

The right-endpoint approximation is shown in Figure

The left-endpoint approximation is
Sketch left-endpoint and right-endpoint approximations for
- Hint
-
Follow the solving strategy in Example
step-by-step.
- Answer
-
The left-endpoint approximation is
. The right-endpoint approximation is . See the below Media.
Looking at Figure
We can demonstrate the improved approximation obtained through smaller intervals with an example. Let’s explore the idea of increasing
The area is approximated by the summed areas of the rectangles, or

Figure

The graph in Figure

We can carry out a similar process for the right-endpoint approximation method. A right-endpoint approximation of the same curve, using four rectangles (Figure

Dividing the region over the interval

Last, the right-endpoint approximation with

Based on these figures and calculations, it appears we are on the right track; the rectangles appear to approximate the area under the curve better as
Value of |
Approximate Area |
Approximate Area |
---|---|---|
Forming Riemann Sums
So far we have been using rectangles to approximate the area under a curve. The heights of these rectangles have been determined by evaluating the function at either the right or left endpoints of the subinterval
A sum of this form is called a Riemann sum, named for the 19th-century mathematician Bernhard Riemann, who developed the idea.
Let
At this point, we'll choose a regular partition
Recall that with the left- and right-endpoint approximations, the estimates seem to get better and better as
Let
Some subtleties here are worth discussing. First, note that taking the limit of a sum is a little different from taking the limit of a function
Second, we must consider what to do if the expression converges to different limits for different choices of
We look at some examples shortly. But, before we do, let’s take a moment and talk about some specific choices for
If we want an overestimate, for example, we can choose
Find a lower sum for
Solution
With
![The graph of f(x) = 10 − x^2 from 0 to 2. It is set up for a right-end approximation of the area bounded by the curve and the x axis on [1, 2], labeled a=x0 to x4. It shows a lower sum.](https://math.libretexts.org/@api/deki/files/4665/5A.png?revision=1&size=bestfit&width=487&height=275)
The Riemann sum is
The area of
- Find an upper sum for
on ; let - Sketch the approximation.
- Hint
-
is decreasing on , so the maximum function values occur at the left endpoints of the subintervals.
- Answer
-
a. Upper sum=
b.
Find a lower sum for
Solution
Let’s first look at the graph in Figure

The intervals are
Using the function
- Hint
-
Follow the steps from Example
.
- Answer
-
Key Concepts
- The use of sigma (summation) notation of the form
is useful for expressing long sums of values in compact form. - For a continuous function defined over an interval
the process of dividing the interval into equal parts, extending a rectangle to the graph of the function, calculating the areas of the series of rectangles, and then summing the areas yields an approximation of the area of that region. - When using a regular partition, the width of each rectangle is
. - Riemann sums are expressions of the form
and can be used to estimate the area under the curve Left- and right-endpoint approximations are special kinds of Riemann sums where the values of are chosen to be the left or right endpoints of the subintervals, respectively. - Riemann sums allow for much flexibility in choosing the set of points
at which the function is evaluated, often with an eye to obtaining a lower sum or an upper sum.
Key Equations
- Properties of Sigma Notation
- Sums and Powers of Integers
- Left-Endpoint Approximation
- Right-Endpoint Approximation
Glossary
- left-endpoint approximation
- an approximation of the area under a curve computed by using the left endpoint of each subinterval to calculate the height of the vertical sides of each rectangle
- lower sum
- a sum obtained by using the minimum value of
on each subinterval
- partition
- a set of points that divides an interval into subintervals
- regular partition
- a partition in which the subintervals all have the same width
- riemann sum
- an estimate of the area under the curve of the form
- right-endpoint approximation
- the right-endpoint approximation is an approximation of the area of the rectangles under a curve using the right endpoint of each subinterval to construct the vertical sides of each rectangle
- sigma notation
- (also, summation notation) the Greek letter sigma (
) indicates addition of the values; the values of the index above and below the sigma indicate where to begin the summation and where to end it
- upper sum
- a sum obtained by using the maximum value of
on each subinterval