6.1 Antiderivatives
Learning Objectives
- 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.
The properties associated with the summation process are given in the following rule.
Rule: Properties of Sigma Notation
Let
Proof
We prove properties 2. and 3. here, and leave proof of the other properties to the Exercises.
2. We have
3. 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.
Rule: Sums and Powers of Integers
- The sum of
integers is given by - The sum of consecutive integers squared is given by
- The sum of consecutive integers cubed is given by
Approximating Area
Now that we have the necessary notation, we return to the problem at hand: approximating the area under a curve. Let

Long description: A graph in quadrant one of an area bounded by a generic curve f(x) at the top, the x-axis at the bottom, the line x = a to the left, and the line x = b to the right. About midway through, the concavity switches from concave down to concave up, and the function starts to increases shortly before the line x = b.
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 Δ
for
Definition
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.
Rule: Left-Endpoint Approximation
On each subinterval

Long description: A diagram showing the left-endpoint approximation of area under a curve. Under a parabola with vertex on the y axis and above the x axis, rectangles are drawn between a=x0 on the origin and b = xn. The rectangles have endpoints at a=x0, x1, x2…x(n-1), and b = xn, spaced equally. The height of each rectangle is determined by the value of the given function at the left endpoint of the rectangle.
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.
Rule: Right-Endpoint Approximation
Construct a rectangle on each subinterval
The notation

Long description: A diagram showing the right-endpoint approximation of area under a curve. Under a parabola with vertex on the y axis and above the x axis, rectangles are drawn between a=x0 on the origin and b = xn. The rectangles have endpoints at a=x0, x1, x2…x(n-1), and b = xn, spaced equally. The height of each rectangle is determined by the value of the given function at the right endpoint of the rectangle.
The graphs in the following figure represent the curve

Long description: Diagrams side by side, showing the differences in approximating the area under a parabolic curve with vertex at the origin between the left endpoints method (the first diagram) and the right endpoints method (the second diagram). In the first diagram, rectangles are drawn at even intervals (delta x) under the curve with heights determined by the value of the function at the left endpoints. In the second diagram, the rectangles are drawn in the same fashion, but with heights determined by the value of the function at the right endpoints. The endpoints in both are spaced equally from the origin to (3, 0), labeled x0 to x6.
In this figure, we draw vertical lines perpendicular to
Looking at (Figure 6.4) and the graphs in (Figure 6.5), we can see that when we use a small number of intervals, neither the left-endpoint approximation nor the right-endpoint approximation is a particularly accurate estimate of the area under the curve. However, it seems logical that if we increase the number of points in our partition, our estimate of A will improve. We will have more rectangles, but each rectangle will be thinner, so we will be able to fit the rectangles to the curve more precisely.
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

Long description: The heights of the rectangles are determined by the values of the function at the left endpoints.
The following figure shows the same curve divided into eight subintervals. Comparing the graph with four rectangles in (Figure 6.6) with this graph with eight rectangles, we can see there appears to be less white space under the curve when

Long description: The heights of the rectangles are determined by the values of the function at the left endpoints.
The next graph shows the same function with 32 rectangles inscribed under the curve. There appears to be little white space left. The area occupied by the rectangles is

Long description: The heights of the rectangles are determined by the values of the function at the left endpoints.
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 6.8), yields an area

Long description: The heights of the rectangles are determined by the values of the function at the right endpoints.
Dividing the region over the interval

Long description: The heights of the rectangles are determined by the values of the function at the right endpoints.
Last, the right-endpoint approximation with

Long description: The heights of the rectangles are determined by the values of the function at the right endpoints.
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
Values of |
Approximate Area Ln | Approximate Area Rn |
---|---|---|
7.5 | 8.5 | |
7.75 | 8.25 | |
7.94 | 8.06 |
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.
Definition
Let
Recall that with the left- and right-endpoint approximations, the estimates seem to get better and better as
Definition
Let
See a graphical demonstration of the construction of a Riemann sum.
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