1.0 Definition

There are several types of area bounded by a curve and the x-

axis. As examples; area bounded by the curve and the x-axis, area

bounded by the curve and the y-axis, the area between two functions

and area bounded by a function whose sign changes.

The area of a region bounded by a graph of a function, the x-

axis, and two vertical boundaries can be determined directly by

evaluating a definite integral. If f(x) ≥ 0 on [ a, b], then the area ( A) of

the region lying below the graph of f(x), above the x-axis, and between

the lines x = a and x = b is

Figure 1

Finding the area under a non-negative function.

If f(x) ≤ 0 on [ a, b], then the area ( A) of the region lying above the

graph of f(x), below the x-axis, and between the lines x = a and x = b is

Figure 2

Finding the area above a negative function.

If f(x) ≥ 0 on [ a, c] and f(x) ≤ 0 on [ c, b], then the area ( A) of the

region bounded by the graph of f(x), the x-axis, and the lines x = a and

x = b would be determined by the following definite integrals:

Figure 3

The area bounded by a function whose sign changes.

Note that in this situation it would be necessary to determine all

points where the graph f(x) crosses the x-axis and the sign of f(x) on

each corresponding interval.

For some problems that ask for the area of regions bounded by

the graphs of two or more functions, it is necessary to determine the

position of each graph relative to the graphs of the other functions of

the region. The points of intersection of the graphs might need to be

found in order to identify the limits of integration. As an example, if f(x)

≥ g( x) on [ a, b], then the area ( A) of the region between the graphs of

f(x) and g( x) and the lines x = a and x = b is

Figure 4

The area between two functions

Note that an analogous discussion could be given for areas

determined by graphs of functions of y, the y-axis, and the lines y = a

and y = b.

Example 1: Find the area of the region bounded by y = x2, the x-axis, x

= –2, and x = 3.

Because f(x) ≥ 0 on [–2,3], the area ( A) is

Example 2: Find the area of the region bounded by y = x3 + x2 – 6 x

and the x-axis.

Setting y = 0 to determine where the graph intersects the x-axis, you

find that

Because f ( x) ≥ 0 on [–3,0] and f ( x) ≤ 0 on [0,2] (see Figure 5 ), the

area ( A) of the region is

Figure 5

Diagram for Example 2.

Example 3: Find the area bounded by y = x2 and y = 8 – x2.

Because y = x2 and y = 8 – x2, you find that

Hence, the curves intersect at (–2,4) and (2,4). Because 8 – x2 ≥ x2 on

[–2,2] (see Figure 6 ), the area ( A) of the region is

Figure 6

Diagram for Example 3.

2.0 Lesson plan for form 6 secondary school in order to find the area

bounded by a curve and the x-axis.

Find the area bounded by the curve and the x-axis.

Learning outcomes: At the end of this unit students will be able to;

Find the area bounded by a curve, the x-axis and the ordinates x = a

and x = b when the area is on one side of the x-axis.

Prior knowledge:

Understand the relationship between the area under a curve and the

definite integral.

Teaching aids: Graph paper.

Procedure:

1. Teacher will explain roughly about area bounded by showing

examples with the aid of slide show of Microsoft power point

software.

a. Sketch the curve.

b. Find the two pints where the curve cuts the x-axis.

c. Apply the area.

2. Teacher will divide students in the group of 5.

3. Students are asked to find the area bounded by the curve

from the given questions by using correct steps.

4. Students will discuss the questions in class.

5. Exercises regarding the topic will be given individually.

6. The teacher will make conclusion of the topic.

Name: ____________________________ Class: ____________

1. Find the area underneath the curve y = x2 + 2 from x = 1 to x = 2.

.

Appendix 1SMK BANDAR BARU SUNGAI BULOH47000 SUNGAI BULOH, SELANGOR.

1. Name: ____________________________

Class: ____________2. Find the area bounded by y = x2 −

4, the x-axis and the lines x = -1 and x =2. .

Name: ____________________________

Class: ____________ 3. What is the area bounded by the

curve y = x3, x = -2 and x = 1? Question

1Solution

Question 2Solution

Question

3SolutionNOTE: In each of Case (1), (2) and (3), the curves

are easy to deal with by summing elements L to R:

We are

(effectively) finding the area by horizontally adding the

areas of the rectangles, width dx and heights y (which we

find by substituting values of x into f(x)).So

(with absolute value signs where

necessary).. Name:

____________________________ Class:

____________Area function1a define A(x) to be the area

bounded by the x-axis and the function f(x) = 3 between the

y-axis and the vertical line at x. ( see the diagram )A(1) =

_________________________A(2) =

_________________________A(3) =

_________________________A(4) =

_________________________And, in general,A(x)

=_______________________________ ( a formula )

Example 1:Step 1:Define A (x) to be the area bounded by

the x-axis and the function f(x) = 1/r between the liner x = 1

and the vertical line at x.Based on your work in problem

1A’(x) = ____________________Computer A(1) =

_______________________Computer A (x) =

______________________Step 2:So the area under f(x) =

1/x between x=1 and x =2 equal to in (2). Outline this area o

the graph. Well use estimates of this area to computer

approximations of In(2).Step 3: Slice the area up in to 4

pieces by drawing 3 evenly spaced vertical lines from the x-

axis up to the curve.Step 4 :Using the left side of each slice

as the height, sketch in 4 rectangles in your graph. What are

the x-coordinates of the sides of the rectangle? Plug these

x-coordinates into f(x) =1/x to computer the heights of the

rectangles. Find the areas of the 4 rectangles and add them

up. This is your first approximation of the area under the

curve, and In(2). Is it an over-estimate or an under-

estimate? Step 5:Using the right side of each slice as the

height, sketch in 4 rectangles in your graph. Find the area of

these rectangles and add them up. This your second

approximation of the area under the curve, and In(2). Is it an

over-estimate or an under-estimate? Step 6:Take the

average of your two estimate to get a new estimate of In(2).

How does it computer with the value given by your

calculator?Step 7:Use the midpoint of each slice to

determine the height and sketch in the resulting 4

rectangles. Use them to approximate In(2). Can you tell if

you are getting an over-estimate or and under-estimate?

Which of your four estimate gives you the closest answers

to the value given by your calculator?3.0 Conduct the

tutorial class to week students on the topic learning

programming. A factory manufactured two types of

souvenirs: toy and key chain, which are made of plastic and

rubber. To make 1 unit of toy, 1 unit plastic and 1 unit of

rubber are needed. For 1 unit of key chain, 5 units of plastic

and 2 units of rubber are needed. The daily amount of

plastic and rubber available for production are 400 and 1100

units respectively. The daily production capacity of factory

for toys and key chains are 150 and 360 respectively. The

net profit obtained from toys and key chains are RM5 and

RM4 per unit respectively. How many of toys and key chains

should be produced by the manufacturer in order to obtain

maximum profit? Formulate this problem as a linear program

to maximized the profits obtained.Solution:Step 1

Understanding the problemStudent

must read the questions carefully until they understand what is the

requirement of questions given.The problem is to determine the

number of toys and key chains to be produced by the manufacturer

in order to obtain maximum profit.Step 2

Identifying all the variablesStudents need to identify or

observe the information given and find the related

information in simple sentences to understand.Consider x as

the number of units of toy to produce and y as the number of

units of key chain to produce.Step 3 Identifying the

objective function and all the constraintsFrom the

information given, x units of plastic will be required to

produce x units of toy and y units of plastic will be required

to produce y units of key chain. So, the total amount of

plastic used to produced to produce x units of toy and y units

of key chain is (x + y) units. The maximum amount of plastic

used per day cannot exceed 400 units.This condition can

expressed algebraically as the inequality x + y 400. This

inequality is known as a constraint for the plastic

resource.Similarly, to produce x units of toy and y units of

key chains, where profit obtained from the sales of toys and

key chains are RM5 and RM4 per unit respectively, the total

amount of rubber required is (5x + 2y) units and this quantity

cannot exceed 1100 units per day. So, the constraint for the

rubber resource is (5x + 2y) 1100.The production capacity

of toys is 150 units per day. This means the factory cannot

manufacture more than 150 units of toy daily. This

requirement is represented by the inequality x

150.Similarly, the production capacity for key chain is

represented by the inequality y 360.The total amount of

profit obtained from selling x units of toy and y units of key

chain is RM (5x + 4y), and this amount is to be

maximized.The variables x and y cannot be negative values

since the represented quantities to be produced. This is

represented by the inequalities x 0 and y 0.Therefore,

the mathematical formulation of the problem is given as

follows:Maximize z = 5x + 4y objective

functionSubject to x + y 400 5x + 2y

1100 x 150

constraint y 360

x 0 , y 0 CI These inequality formulas will

ease the students to clarify the questions

given.REFERENCESAisah Ali, Nor Hayati Md. Yusof,

HBMT4403, Teaching of Mathematics for Form Six

Secondary School, July 2009, Open University Malaysia,

Kuala Lumpur.David F Treagust, Reinders Devit, Barry J

Fraser, (1996). Improving Teaching and Learning in

Sciences and Mathematics, Teachers College Press, New

York.Lilian C M c Dermott, In Electronic Journal Of

Mathematics Education, Vol 2, No 2,- December 1997.Tey

Kim Soon, Goh Choon Booy, Tan Ah Geok, Matematik STPM,

Sukatan S & T, 1995, Penerbitan Pelangi Sdn. Bhd, Johor

Bahru.http://www.cliffsnotes.com/WileyCDA/CliffsReviewTopic/

topicArticleId-39909,articleId-39905.html>.http://

www.intmath.com/Applications-integration/2_Area-under-

curve.php