PPT: Application of Integration | Engineering Mathematics - Civil Engineering (CE) PDF Download

 Page 1


Applications Of The Definite Integral
Applications Of The Definite Integral
The Area under the curve of a function
The area between two curves
The Volume of the Solid of revolution
?
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 ? f(x)          . 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.
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Page 2


Applications Of The Definite Integral
Applications Of The Definite Integral
The Area under the curve of a function
The area between two curves
The Volume of the Solid of revolution
?
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 ? f(x)          . 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.
© iTutor. 2000-2013. All Rights Reserved
Area under a Curve
Area under a Curve
[ ] ) ( ) ( ) ( ) ( a F b F x F dx x f
b
a
b
a
- = =
?
To 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 :
y= f(x)
Area = 
?
b
a
dx x f ) (
x
Y
x = a x= b
© iTutor. 2000-2013. All Rights Reserved
Page 3


Applications Of The Definite Integral
Applications Of The Definite Integral
The Area under the curve of a function
The area between two curves
The Volume of the Solid of revolution
?
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 ? f(x)          . 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.
© iTutor. 2000-2013. All Rights Reserved
Area under a Curve
Area under a Curve
[ ] ) ( ) ( ) ( ) ( a F b F x F dx x f
b
a
b
a
- = =
?
To 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 :
y= f(x)
Area = 
?
b
a
dx x f ) (
x
Y
x = a x= b
© iTutor. 2000-2013. All Rights Reserved
But  when the graph line is below the ‘x’ axis, the definite 
integral is negative. The area is then given by:
y= f(x)
Area = 
?
- b
a
dx x f ) (
© iTutor. 2000-2013. All Rights Reserved
Page 4


Applications Of The Definite Integral
Applications Of The Definite Integral
The Area under the curve of a function
The area between two curves
The Volume of the Solid of revolution
?
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 ? f(x)          . 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.
© iTutor. 2000-2013. All Rights Reserved
Area under a Curve
Area under a Curve
[ ] ) ( ) ( ) ( ) ( a F b F x F dx x f
b
a
b
a
- = =
?
To 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 :
y= f(x)
Area = 
?
b
a
dx x f ) (
x
Y
x = a x= b
© iTutor. 2000-2013. All Rights Reserved
But  when the graph line is below the ‘x’ axis, the definite 
integral is negative. The area is then given by:
y= f(x)
Area = 
?
- b
a
dx x f ) (
© iTutor. 2000-2013. All Rights Reserved
(Positive)
 
(Negative)
2
1
0
2
1
2
1
1
0
1
0
2
= - =
?
?
?
?
?
?
=
?
x xdx
1 - 1
1 1 - 2
1
2
1
0
2
1
0
1
0
1
2
- = - =
?
?
?
?
?
?
=
- - ?
x xdx
© iTutor. 2000-2013. All Rights Reserved
Page 5


Applications Of The Definite Integral
Applications Of The Definite Integral
The Area under the curve of a function
The area between two curves
The Volume of the Solid of revolution
?
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 ? f(x)          . 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.
© iTutor. 2000-2013. All Rights Reserved
Area under a Curve
Area under a Curve
[ ] ) ( ) ( ) ( ) ( a F b F x F dx x f
b
a
b
a
- = =
?
To 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 :
y= f(x)
Area = 
?
b
a
dx x f ) (
x
Y
x = a x= b
© iTutor. 2000-2013. All Rights Reserved
But  when the graph line is below the ‘x’ axis, the definite 
integral is negative. The area is then given by:
y= f(x)
Area = 
?
- b
a
dx x f ) (
© iTutor. 2000-2013. All Rights Reserved
(Positive)
 
(Negative)
2
1
0
2
1
2
1
1
0
1
0
2
= - =
?
?
?
?
?
?
=
?
x xdx
1 - 1
1 1 - 2
1
2
1
0
2
1
0
1
0
1
2
- = - =
?
?
?
?
?
?
=
- - ?
x xdx
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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
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