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Area Under A Curve –– Riemann Sums

Author: Sophia
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what's covered
In this lesson, you will form Riemann sums to approximate areas. This idea is very important as it paves the way for some applications in integral calculus. Specifically, this lesson will cover:

Table of Contents

1. Definition of Riemann Sum

Suppose we want to calculate the area between the graph of a nonnegative function f open parentheses x close parentheses and the x-axis interval open square brackets a comma space b close square brackets comma as shown in the figure below.

A graph with an x-axis and a y-axis represents the function y equals f(x). The x-axis has two labeled points ‘a’ and ‘b’, where b is greater than a, defining an interval. A curve opens upward, begins from the upper left part of the graph, dips downward, and then rises upward toward the upper right. Two dashed vertical lines at x equals a and x equals b, intersect the curve at two different points. The area under the curve and above the horizontal axis between x equals a and x equals b is shaded.

If f open parentheses x close parentheses is nonnegative, the Riemann sum method is to build several rectangles with bases on the interval open square brackets a comma space b close square brackets and sides that reach up to the graph of f open parentheses x close parentheses. Then, the areas of the rectangles can be calculated and added together to get a number called a Riemann sum of f open parentheses x close parentheses on open square brackets a comma space b close square brackets.

A graph with an x-axis and a y-axis represents the function y equals f(x) with an increasing curve. The x-axis has points labeled x sub 0, x sub 1, x sub 2, and x sub 3, representing intervals. Three rectangular bars correspond to the subintervals x sub 0 – x sub 1, x sub 1 – x sub 2, and x sub 2 – x sub 3, respectively. The bars increase in height as ‘x’ increases. A curve rises upward from the left of the first bar up to the second, dips downward slightly, and then curves upward slightly and rises. The curve touches each bar at x equals x sub 0, x equals x sub 1, and x equals x sub 2, forming an ascending stair-like pattern. The total area of the rectangular bars is shaded.

The area of the region formed by the rectangles is an approximation of the area between the graph and the x-axis.

term to know
Riemann Sum
The sum obtained from the areas of rectangles that are used to approximate the area between a curve and the x-axis.

2. Finding the Riemann Sum

In order to find the Riemann sum, there are several quantities that need to be established first.

  1. Find the partition and subintervals.
  2. Find the width of each subinterval.
  3. Select x-values within each partition.
  4. Form the Riemann sum.
Let’s take a deeper look at each step.

2a. Find the Partition and Subintervals

First, a partition of the interval open square brackets a comma space b close square brackets is needed to establish the bases of the rectangles. Consider the graph in the figure.

A graph with an x-axis and a y-axis represents the function y equals f(x) with an increasing curve. The x-axis has points labeled x sub 0, x sub 1, x sub 2, and x sub 3, representing intervals. Three rectangular bars correspond to the subintervals x sub 0 – x sub 1, x sub 1 – x sub 2, and x sub 2 – x sub 3, respectively. The bars increase in height as ‘x’ increases. A curve rises upward from the left of the first bar up to the second, dips downward slightly, and then curves upward slightly and rises. The curve touches each bar at x equals x sub 0, x equals x sub 1, and x equals x sub 2, forming an ascending stair-like pattern. The total area of the rectangular bars is shaded.

The endpoints of the interval are x subscript 0 and x subscript 3. In order to form three rectangles, two more values
(x subscript 1 and x subscript 2) are added to form a partition of the interval open square brackets x subscript 0 comma space x subscript 3 close square brackets.

We label the partition by the x-coordinates, namely open curly brackets x subscript 0 comma space x subscript 1 comma space x subscript 2 comma space x subscript 3 close curly brackets. The numbers are listed in increasing order.

Note that there are 4 x-values in the partition for three rectangles. In general, if n rectangles are desired, there would be n + 1 x-values in the partition. This is why the first one is labeled as x subscript 0 (which is the left-hand endpoint of the interval), so the last one can be called x subscript n (to match the number of rectangles).

The subintervals for this partition are open square brackets x subscript 0 comma space x subscript 1 close square brackets comma open square brackets x subscript 1 comma space x subscript 2 close square brackets comma and open square brackets x subscript 2 comma space x subscript 3 close square brackets.

terms to know
Partition
A set of x-values that are used to split the interval open square brackets a comma space b close square brackets into smaller intervals.
Subinterval
A smaller interval that is part of a larger interval.

2b. Find the Width of Each Subinterval

It is most convenient to select a partition where each x-value is the same distance apart from its neighbor, but that is not necessary.

Continuing with this partition, we use the notation increment x subscript k to represent the width of the k to the power of t h end exponent subinterval. Recall that the width of an interval is the difference between its endpoints.

Subinterval Width
open square brackets x subscript 0 comma space x subscript 1 close square brackets increment x subscript 1 equals x subscript 1 minus x subscript 0
open square brackets x subscript 1 comma space x subscript 2 close square brackets increment x subscript 2 equals x subscript 2 minus x subscript 1
open square brackets x subscript 2 comma space x subscript 3 close square brackets increment x subscript 3 equals x subscript 3 minus x subscript 2

2c. Select x-Values Within Each Partition

Let c subscript k equals the value of x used in the k to the power of t h end exponent subinterval. There are popular choices for c subscript k colon

  • The left endpoint of each subinterval
  • The right endpoint of each subinterval
  • The midpoint of each subinterval
Of course, we are not forced to use any one of these, but these are the most convenient.

2d. Form the Riemann Sum

Consider the figure shown below:

A graph with an x-axis and a y-axis represents the function y equals f(x) with an increasing curve. The x-axis has points labeled x sub 0, x sub 1, x sub 2, and x sub 3, representing intervals. Three rectangular bars correspond to the subintervals x sub 0 – x sub 1, x sub 1 – x sub 2, and x sub 2 - x sub 3, respectively. The bars increase in height as ‘x’ increases. A curve rises upward from the left of the first bar up to the second, dips downward slightly, and then curves upward slightly and rises. The curve touches each bar at x equals x sub 0, x equals x sub 1, and x equals x sub 2, forming an ascending stair-like pattern. The total area of the rectangular bars is shaded.

As the rectangles suggest, the left-hand endpoint was used in each sub-interval to set the height of the rectangle. This means:

Subinterval Value Chosen Width
open square brackets x subscript 0 comma space x subscript 1 close square brackets c subscript 1 equals x subscript 0 increment x subscript 1 equals x subscript 1 minus x subscript 0
open square brackets x subscript 1 comma space x subscript 2 close square brackets c subscript 2 equals x subscript 1 increment x subscript 2 equals x subscript 2 minus x subscript 1
open square brackets x subscript 2 comma space x subscript 3 close square brackets c subscript 3 equals x subscript 2 increment x subscript 3 equals x subscript 3 minus x subscript 2

So, we can say:

  • Area of the first rectangle: f open parentheses c subscript 1 close parentheses times increment x subscript 1
  • Area of the second rectangle: f open parentheses c subscript 2 close parentheses times increment x subscript 2
  • Area of the third rectangle: f open parentheses c subscript 3 close parentheses times increment x subscript 3
Then, the approximation for the area between f open parentheses x close parentheses and the x-axis is the sum of these areas. Written using sigma notation, the Riemann sum is:

sum from k equals 1 to 3 of f open parentheses c subscript k close parentheses times increment x subscript k

In general, here is the definition (formula) for a Riemann sum.

formula to know
Riemann Sum
When approximating the area between a nonnegative function y equals f open parentheses x close parentheses and the x-axis by using n rectangles, the summation sum from k equals 1 to n of f open parentheses c subscript k close parentheses times increment x subscript k is called the Riemann sum, where c subscript k is a value of x in the k to the power of t h end exponent subinterval, and increment x subscript k is the width of the k to the power of t h end exponent subinterval.

3. Using Riemann Sums to Calculate Area

Now that we have all the definitions, let’s compute a few Riemann sums.

EXAMPLE

Use a Riemann sum with 4 rectangles of equal width to approximate the area between y equals x squared plus 1 and the x-axis on the interval open square brackets 1 comma space 3 close square brackets. Use the left-hand endpoint of each subinterval.

Since each subinterval will have equal width, that width is fraction numerator w i d t h space o f space open square brackets 1 comma space 3 close square brackets over denominator 4 end fraction equals 2 over 4 equals 0.5.

Based on the problem, we have the following information:

Subinterval Width of Subinterval Value Chosen in Each Subinterval
open square brackets 1 comma space 1.5 close square brackets 0.5 1
open square brackets 1.5 comma space 2 close square brackets 0.5 1.5
open square brackets 2 comma space 2.5 close square brackets 0.5 2
open square brackets 2.5 comma space 3 close square brackets 0.5 2.5

Here's a picture of the graph with the rectangles that were used:

A graph with an x-axis ranging from 0 to 3 at intervals of 0.5 and a y-axis ranging from 0 to 8. Four rectangular bars correspond to subintervals 1–1.5, 1.5–2, 2–2.5, and 2.5–3, respectively, on the x-axis, increasing in height as x increases. A parabolic curve starts from the left of the y-axis; reaches an inverted peak at (0, 1); rises upward from x equals 0.3; touches the top left corner of each bar at (1, 2), (1.5, 3.25), (2, 5), and (2.5, 7.25); and rises upward beyond x equals 2.5, forming an ascending stair-like pattern. The total area of the rectangular bars is shaded.

Then, the Riemann sum is:

sum from k equals 1 to 4 of f open parentheses c subscript k close parentheses increment x subscript k equals f open parentheses 1 close parentheses times 0.5 plus f open parentheses 1.5 close parentheses times 0.5 plus f open parentheses 2 close parentheses times 0.5 plus f open parentheses 2.5 close parentheses times 0.5 Use the Riemann sum formula.
equals 0.5 open square brackets f open parentheses 1 close parentheses plus f open parentheses 1.5 close parentheses plus f open parentheses 2 close parentheses plus f open parentheses 2.5 close parentheses close square brackets Factor out 0.5.
equals 0.5 open parentheses 2 plus 3.25 plus 5 plus 7.25 close parentheses Substitute values: f open parentheses 1 close parentheses equals 2 comma f open parentheses 1.5 close parentheses equals 3.25 comma f open parentheses 2 close parentheses equals 5 comma f open parentheses 2.5 close parentheses equals 7.25
equals 8.75 Simplify.

Thus, an approximation of the area is 8.75 space units squared.

big idea
When the width of each subinterval is the same, we call the width of the interval increment x since they are all the same, then increment x equals fraction numerator b minus a over denominator n end fraction.

try it
Use a Riemann sum with 4 rectangles of equal width to approximate the area between y equals x squared plus 1 and the x-axis on the interval open square brackets 1 comma space 3 close square brackets. Use the right-hand endpoint of each subinterval. Note, this is the same information as in the last example, except that right-hand endpoints are used.
Approximate the area.
Refer to the figure in the last example. The right-hand x-values are 1.5, 2, 2.5, and 3.

Recall that the width of each rectangle is 0.5 (side along the x-axis).

From the function f open parentheses x close parentheses equals x squared plus 1 comma we have f open parentheses 1.5 close parentheses equals 3.25 comma f open parentheses 2 close parentheses equals 5 comma f open parentheses 2.5 close parentheses equals 7.25 comma and f open parentheses 3 close parentheses equals 10.

Then, the right-hand area estimate is:

0.5 open square brackets f open parentheses 1.5 close parentheses plus f open parentheses 2 close parentheses plus f open parentheses 2.5 close parentheses plus f open parentheses 3 close parentheses close square brackets equals 0.5 open square brackets 3.25 plus 5 plus 7.25 plus 10 close square brackets</dd></dl> space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 0.5 open square brackets 25.5 close square brackets space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 12.75 space units squared

Let’s look at an example where the widths of the intervals are not the same.

EXAMPLE

Consider the function f open parentheses x close parentheses equals square root of x on the interval open square brackets 0 comma space 25 close square brackets. Estimate the area between f open parentheses x close parentheses and the x-axis by using the partition open curly brackets 0 comma space 1 comma space 4 comma space 9 comma space 16 comma space 25 close curly brackets. Use the right-hand endpoint of each subinterval.

Since there are 6 numbers in the partition, there are 5 rectangles. The table shows the information we need to set this up:

Subinterval Width of Subinterval Value Chosen in Each Subinterval
open square brackets 0 comma space 1 close square brackets 1 1
open square brackets 1 comma space 4 close square brackets 3 4
open square brackets 4 comma space 9 close square brackets 5 9
open square brackets 9 comma space 16 close square brackets 7 16
open square brackets 16 comma space 25 close square brackets 9 25

The graph of f open parentheses x close parentheses along with the rectangles is shown below.

A graph with an x-axis ranging from 0 to 25 and a y-axis ranging from 0 to 5. Five rectangular bars correspond to subintervals 0–1, 1–4, 4–9, 9–16, and 16–25, respectively, on the x-axis, increasing in height as x increases. A sideways parabolic curve starts from the origin (0, 0); then gradually rises upward; touches the top right corners of each rectangular bar at (1, 1), (4, 2), (9, 3), (16, 4), and (25, 5); and extends beyond x equals 30. The total area of the rectangular bars is shaded.

Then, the Riemann sum is sum from k equals 1 to 5 of f open parentheses c subscript k close parentheses increment x subscript k.

sum from k equals 1 to 5 of f open parentheses c subscript k close parentheses increment x subscript k equals f open parentheses 1 close parentheses times 1 plus f open parentheses 4 close parentheses times 3 plus f open parentheses 9 close parentheses times 5 plus f open parentheses 16 close parentheses times 7 plus f open parentheses 25 close parentheses times 9 Use the Riemann sum formula.
equals 1 open parentheses 1 close parentheses plus 2 open parentheses 3 close parentheses plus 3 open parentheses 5 close parentheses plus 4 open parentheses 7 close parentheses plus 5 open parentheses 9 close parentheses Substitute values: f open parentheses 1 close parentheses equals 1 comma f open parentheses 4 close parentheses equals 2 comma f open parentheses 9 close parentheses equals 3 comma f open parentheses 16 close parentheses equals 4 comma f open parentheses 25 close parentheses equals 5
equals 95 Simplify.

Thus, the approximation for the area is 95 space units squared.

watch
In this video, we will use a Riemann sum to approximate the area below the graph of f open parentheses x close parentheses equals x squared plus 2 on the interval open square brackets 0 comma space 4 close square brackets using 4 rectangles of equal width, using the left-hand endpoints of each subinterval.

think about it
What effect would increasing the number of rectangles (partitions) have on the estimate in terms of the actual area?

summary
In this lesson, you learned that a Riemann sum provides a systematic way to approximate the area between a curve y equals f open parentheses x close parentheses and the x-axis on the interval open square brackets a comma space b close square brackets, by obtaining the sum from the areas of rectangles. You learned that when finding the Riemann sum, there are several quantities that need to be established first: find the partition and subintervals; find the width of each subinterval; select x-values within each partition; and finally, form the Riemann sum. Using this knowledge, you were then able to explore several examples of using Riemann sums to calculate area. Many applications we will investigate later in this course are based on Riemann sums, which makes this a very important topic to understand.

Source: THIS TUTORIAL HAS BEEN ADAPTED FROM CHAPTER 4 OF "CONTEMPORARY CALCULUS" BY DALE HOFFMAN. ACCESS FOR FREE AT WWW.CONTEMPORARYCALCULUS.COM. LICENSE: CREATIVE COMMONS ATTRIBUTION 3.0 UNITED STATES.

Terms to Know
Partition

A set of x-values that are used to split the interval open square brackets a comma space b close square brackets into smaller intervals.

Riemann Sum

The sum obtained from the areas of rectangles that are used to approximate the area between a curve and the x-axis.

Subinterval

A smaller interval that is part of a larger interval.

Formulas to Know
Riemann Sum

When approximating the area between a nonnegative function y equals f open parentheses x close parentheses and the x-axis by using n rectangles, the summation sum from k equals 1 to n of f open parentheses c subscript k close parentheses times increment x subscript k is called the Riemann Sum, where c subscript k is a value of x in the k to the power of t h end exponent subinterval, and increment x subscript k is the width of the k to the power of t h end exponent subinterval.

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