applications of the definite integral the area under the curve of a function the area between two...
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Applications Of The Definite IntegralApplications Of The Definite Integral
The Area under the curve of a function
The area between two curves
The Volume of the Solid of revolution
Applications of the Definite Integral
• In calculus, the integral of a function is an extension of the concept of a sum. The process of finding integrals is called integration. The process is usually used to find a measure of totality such as area, volume, mass, displacement, etc.
The integral would be written . The∫ sign represents integration, a and b are the endpoints of the interval, f(x) is the function we are integrating known as the integrand, and dx is a notation for the variable of integration. Integrals discussed in this project are termed definite integrals.
We use definite integrals
• basically,
• Results:
Given:
evaluate,
Solution:
Area under a CurveTo find the area under a curve. This expression gives us a
definite value (a number) at the end of the calculation.
When the curve is above the ‘x’ axis, the area is the same as the definite integral :
But when the graph line is below the ‘x’ axis, the definite integral is negative. The area is then given by:
(Positive)
(Negative)
Example 1:
let f (x)=2-x .Find the area bounded by the curve of f , the x-axis and the lines x=a and x=b for each of the following cases:
a=-2 b=2
a=2 b=3
a=-2 b=3
The graph:
Is a straight line y=2-x:
F (x) is positive on the interval [-2, 2)
F (x) is negative on the interval (2, 3]
2
2 3-2
Case 1:
The area A1 between f, the x-axis and the
lines x=-2 and x=2 is: f(x)>0; x [-2,2)
862
)2
44()
2
44(
)2
2(
)2(
2
)(
2
2
2
2
2
2
2
2
2
1
xx
dxx
dxx
dxxfA
2
32-2
A1
f(x)<0; x (2, 3]
Case2
The area A2 between f, the x-axis and the
Lines x=2 and x=3 is:
2/1
)2
44()
2
96(
)2
2()2(
2
)(
32
23
2
3
2
3
2
1
dxx
xdxx
dxx
dxxfA
3
2
2-2
Case3: The area a between f, the X-axis and the lines
X=-2 and X=3 is :
2/17
2/18
)2()2(
2
)(
3
2
2
2
3
2
3
2
dxxdxx
dxx
dxxf
2
2
3-2
Area Bounded by 2 Curves
Say you have 2 curves y = f(x) and y = g(x)
Area under f(x) =
Area under g(x) =
Superimposing the two graphs, Area bounded by f(x) and g(x)
Example (2)
Let f (x) =X, g (X) = Find the area between f and g from X=a to X=b Following cases
a=-1 b=0
a=0 b=1
a=-1 b=1
g (X)>f (X) on (-1,0) and hence on this interval
, we have: g (X) –f (X)>0
So |g (X) –f (X)| =g (X)-f (X)= -x 5x
5x
Case (1):The area A1 between f and g from X= -1 and x=0
is: is:g (X)>f (X) on (-1,0) and hence on this interval
, we have :
g (X) –f (X)>0
So |g (X) –f (X)| =g (X)-f (X)= -x
3/1
)2/16/1()00(
)2/6/(
)(
)()(
0
1
26
50
1
0
1
1
xx
dxxx
dxxfxgA1
1
11
g
5x
Case (2)
The area A between f and g from X = 0 to X=1f(x) >g (X) on(0,1) and hence on this interval
, we have
F(X) –g (X)>0 so |g (X) –f (X)| =f (X) –g (X) =x-
3
100)
6
1
2
1(
6/2/
)()(
1
0
62
1
0
5
1
0
2
xx
dxxx
dxXfXgA
1
1
11
gf
5x
Case (3)
The area A between f and g from X = -1 to X=1
3/2
3/13/1
)()(
)()(
1
0
550
1
1
1
3
dxxxxx
dxxfxgA
1
1
11
gf
Volumes of Revolution :V=Π∫ (x)dx
• A solid of revolution is formed when a region bounded by part of a curve is rotated about a straight line.
2f
Rotation about x-axis:
Rotation about y-axis:
Example: Find the volume of the solid generated by revolving the region bounded by the graph of
y = x, y = 0, x = 0 and x = 2. At the solid
Solution: we shall now use definite integrals to find the volume defined above. If we let
f(x) = x according to 1 above, the volume is given by the definite integral
Volume
3/8
]3/[(
)(
20
3
2
0
2
2
0
2
22
1
x
dxx
dxx
dxxfx
x
Example1:1
Consider the area bounded by the graph of the function
f(x) =x- and x-axis:
The volume of solid is:
30/
)5/04/03/0()5/14/23/1(
)5/23/(
)2(
)(
1
0
533
431
0
2
1
0
22
xxx
dxxxx
dxxx
1
2x
In conclusion, an integral is a mathematical object that can be interpreted as an area or a
generalization of area. Integrals, together with derivatives, are the fundamental objects of calculus. Other words for integral include
antiderivative and primitive.
The group members:
• Sabrina Kamal ___________________ID:2004/58527• Manal Alsaadi ____________________ID:2004/51562• Taiba Mustafa ____________________ID:2005/50524• Muneera Ahmed__________________ID:2004/550244
Math119 - Section (1)Fall 2006
Dr.F.K.Al-Muhannadi