Unit 8: Area Between Curves and Applications of Integration

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Unit 8: Area Between Curves

|DAY |TOPIC |ASSIGNMENT |

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|1 |Area Between Two Curves |p. 9-10 |

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|2 |Area Between Two Curves |p. 11-12 (Worksheet) |

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|3 |Area Between Two Curves |p. 13-14 (Worksheet) |

| | | |

|4 |Quiz | |

| | | |

Area Between Two Curves

Learning Objectives

A student will be able to:

• Compute the area between two curves with respect to the [pic]and [pic]axes.

In the last chapter, we introduced the definite integral to find the area between a curve and the [pic]axis over an interval [pic]In this lesson, we will show how to calculate the area between two curves.

Consider the region bounded by the graphs [pic]and [pic]between [pic]and [pic]as shown in the figures below. If the two graphs lie above the [pic]axis, we can interpret the area that is sandwiched between them as the area under the graph of [pic]subtracted from the area under the graph [pic]

[pic]

 

[pic]

 

[pic]

 

Therefore, as the graphs show, it makes sense to say that

[Area under [pic](Fig. 1a)] [pic][Area under [pic](Fig. 1b)] [pic][Area between [pic]and [pic](Fig. 1c)],

[pic]

This relation is valid as long as the two functions are continuous and the upper function [pic]on the interval [pic]

The Area Between Two Curves (With respect to the [pic]axis)

If [pic]and [pic]are two continuous functions on the interval [pic]and [pic]for all values of [pic]in the interval, then the area of the region that is bounded by the two functions is given by

[pic]

Example 1:

Find the area of the region enclosed between [pic]and [pic]

[pic]

 

Solution:

We first make a sketch of the region (Figure 2) and find the end points of the region. To do so, we simply equate the two functions,

[pic]

and then solve for [pic]

[pic]

from which we get [pic]and [pic]

So the upper and lower boundaries intersect at points [pic]and [pic]

As you can see from the graph, [pic]and hence [pic]and [pic]in the interval [pic]Applying the area formula,

[pic]

Integrating,

[pic]

So the area between the two curves [pic]and [pic]is [pic]

Sometimes it is possible to apply the area formula with respect to the [pic]coordinates instead of the [pic]coordinates. In this case, the equations of the boundaries will be written in such a way that [pic]is expressed explicitly as a function of [pic](Figure 3).

[pic]

 

The Area Between Two Curves (With respect to the [pic]axis)

If [pic]and [pic]are two continuous functions on the interval [pic]and [pic]for all values of [pic]in the interval, then the area of the region that is bounded by [pic]on the left, [pic]on the right, below by [pic]and above by [pic]is given by

[pic]

Example 2:

Find the area of the region enclosed by [pic]and [pic]

Solution:

[pic]

 

As you can see from Figure 4, the left boundary is [pic]and the right boundary is [pic]The region extends over the interval [pic]However, we must express the equations in terms of [pic]We rewrite

[pic]

Thus

[pic]

Multimedia Links

For a video presentation of the area between two graphs (14.0)(16.0), see Math Video Tutorials by James Sousa, Area Between Two Graphs (6:12)[pic].

For an additional video presentation of the area between two curves (14.0)(16.0), see Just Math Tutoring, Finding Areas Between Curves (9:50)[pic].

Review Questions

In problems #1 - 7, sketch the region enclosed by the curves and find the area.

1. [pic] on the interval [pic]

2. [pic] [pic] on the interval [pic]

3. [pic] [pic]

4. [pic] [pic] [0, 2(]

5. [pic] [pic] integrate with respect to y

6. [pic] [pic]

7. [pic] [pic] [pic]

8. Find the area enclosed by [pic] and [pic]

9. If the area enclosed by the two functions [pic] and [pic] is 2, what is the value of k?

10. Find the horizontal line y = k that divides the region between [pic] and [pic] into two equal areas.

Review Answers

1. [pic]

2. [pic]

3. [pic]

4. [pic]

5. [pic]

6. [pic]

7. [pic]

8. [pic]

9. [pic]

10. [pic]

Area Between Two Curves Practice

Sketch the region bounded by the graphs of the functions and find the area of the region. You may use fnInt to calculate the area, but graph without the aide of your calculator!

1. [pic] 2. [pic]

3. [pic] 4. [pic]

5. [pic] 6. [pic]

7. [pic] 8. [pic]

9. [pic] 10. [pic]

Answers:

1. [pic] 2. [pic] 3. [pic] 4. [pic] 5. 2

6. [pic] 7. [pic] 8. [pic] 9. 9 10. [pic]

Area Between 2 Curves Practice

Find the area between the curves.

1.) [pic]

2.) [pic]

3.) [pic]

4.) [pic]

5.) [pic]

Answers:

|1.) [pic] |4.) [pic] |

| | |

|2.) [pic] |5.) [pic] |

| | |

|3.) [pic] | |

Day 1

Find the area bounded by the graphs of the functions. Remember to sketch the graphs if they are not already given and find the intersections first then write and evaluate your integral.

1. [pic]

2. [pic]

3. [pic]

4. [pic]

Day 2

Calculus X Name_______________________________

More Area Between Curves WS

As per the new usual, I will be collecting this. It is worth 10 points, 7 for completion and 3 for the correctness of 3 randomly chosen problems.

Find the area bounded by the graphs of the functions. Remember to sketch the graphs and find the intersections first then write and evaluate your integral.

1. [pic]

2. [pic]

3. [pic]

4. [pic]

5. [pic]

6. [pic]

7. [pic]

Day 3

Calculus X Name_______________________________

Area Between Curves WS

Find the area bounded by the graphs of the functions. Remember to sketch the graphs and find the intersections first then write and evaluate your integral.

1. [pic]

2. [pic]

3. [pic]

4. [pic]

5. [pic]

6. [pic]

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