Examville is a global education community where users like you can connect and interact with other students and teachers from around the world Share seek download and discuss everything inside an

Document Sample

Examville is a global education community where users like you can
connect and interact with other students and teachers from around
outside the classroom.

All you need is an email address and a password to get started.

APPLICATIONS OF THE DEFINITE INTEGRAL
The definite integral of a function are used in many applications. Those discussed here
are
•       Areas bounded by curves,
•       Volumes
•       Lengths of arcs of curves.

AREAS
The area of a region bounded by the graph of a function, the x-axis, and two vertical lines
can be done by using definite integral. If in an interval [a,b], f x ≥ 0 , then the region
` a

` a
lying below the graph of f x , above the x-axis and between the two vertical lines x = a
and x = b is given by
b

A = Z f x dx
` a

a

Figure: Finding the area under a positive function
If f x ≤ 0 on [a,b], then the area of the region lying above the graph of f x , below
` a                                                                           ` a

the x-axis and between the two vertical lines x = a and x = b is given by
L         M
b               L b
L
M
M
A = @ Z f x dx =              L Z f x dxM
` a         L    ` a M
L         M
L         M
a               La        M

Figure : Finding the area above a negative function

Note that in both the cases the graph does not intersect the x-axis.

If the graph of f x intersects x-axis at x = c, where a < c < b , and if f x ≥ 0 on [a,c]
` a                                                 ` a

and f x ≤ 0 on [c,b] , then the area of the region bounded by the graph of f x , the x-
` a                                                                        ` a

axis and the two vertical lines x = a and x = b is given by
b                    c            b

A =                      = Z f x dx @ Z f x dx
L ` a M
Z L f x Mdx
` a          ` a
L     M
a                      a             c
Figure : The area bounded by a function whose sign changes.

In such situations it is necessary to determine all the points where the function changes
sign , that is , the points where the curve intersects the x-axis, and to determine the sign of
f(x) in each interval.

We now move on to find areas of regions bounded by the graphs of two or more
functions, or two functions and two vertical lines x = a and x = b . First, we need to
locate the position of each graph relative to the position of the other. The points of
intersection of the graphs might be necessary.

For example, if f x ≥ g x
` a    ` a
at all points on [a,b] , then the area between the graphs of
f(x) and g(x) and two vertical lines x = a and x = b is given by
b

A = Z f x @ g x dx
B ` a   ` aC

a
Figure : The area between two functions.

Area with the y-axis :
x = f y , the y-axis and between the
` a
We can also find the area between graph of
horizontal lines y = a and y = b.
If in an interval [a,b], f y ≥ 0 , then the region lying on the left of the graph of f y ,
` a                                                         ` a

on the right of the y-axis and between the two horizontal lines y = a and y = b is given
by
b

A = Z f y dy
` a

a
Similarly, in an interval [a,b], f y ≤ 0 , then the region lying on the right of the graph
` a

` a
of f y , on the left of the y-axis and between the two horizontal lines y = a and y = b
is given by
b

A = @ Z f y dy
` a

a

Example : Find the area of the region bounded by
y = x 2 , the x @ axis, x = @ 3 and x = 2 A
Solution :
Here f x = x 2 and f x ≥ 0 in the interval @ 3, 2
` a               ` a                    B     C

2

So, the area        A = Z x 2 dx
@3
1f C2
f
= fx 3
B
3 @3
1f 3 ` a3C
f
= f2 @ @ 3
B
3
35f
ff
= fff
f
3
Example : Find the area of the region bounded by the curve y = x 3 + x 2 @ 6x and the x-
axis.
Solution : Below is the graph of the function.
Here y = f x = x 3 + x 2 @ 2x
` a

To know where the curve intersects the x @ axis, we put y = 0 in the equation,
x 3 + x 2 @ 2x = 0,
b            c
x x 2 + x @ 2 = 0,
x x + 2 x @1 =0
`     a`            a

x = 0, x = @ 2, x = 1
B     C   B   C
So the curve bounds an area with the x @ axis in the interval @ 2,0 and 0,1 A

f x ≥ 0 , and in the interval 0,1               f x ≤0
B       C     ` a                                B       C   ` a
In the interval @ 2,0
1

A = Z x 3 + x 2 @ 2x dx
B                C

@2
0                             1

= Z x 3 + x 2 @ 2x dx @ Z x 3 + x 2 @ 2x dx
B                C            B              C

@2                                0
0                               1
F 1f
ff
f      1f
ff                             F 1f
ff
f  1f
ff
=           x + x3@ x2                            x + x3@ x2
G                                     G
4                                    4
@
4    3                                4    3
@2                              0

8f F
f
f              5f
ff
ff
ff
=
G
@ @
3               12
37f
f
ff
ff
f      1f
ff
ff
ff
=        = 3
12      12

Example : Find the area bounded by the curves y = x 2 @ 6 and y = 12 @ x 2 .

Solution :
To find the points

DOCUMENT INFO
Shared By:
Categories:
Stats:
 views: 79 posted: 3/24/2010 language: English pages: 13
Description: Calculus study notes from Examville - Definite Integrals
BUY THIS DOCUMENT NOW PRICE: \$6.95 100% MONEY BACK GUARANTEED
PARTNER
Examville is a global education community where users can connect and interact with other students and teachers from around the world. Share, seek, download and discuss everything inside and outside the classroom. Visit us at http://www.examville.com