4029 - Properties of the Definite Integral
• Think rectangles A  bh
d  rt
• Distance
a

A).
a
f ( x)dx  0 _____________________________________
b
 1dx  (b  a) _____________________________________
B).
C) .
D).
E).
a
b

a

b
a

b
a
a
f ( x)dx    f ( x)dx
b
______________________________________________
b
kf ( x)dx  k  f ( x)dx
a
________________________________________________
b
b
a
a
( f ( x)  g ( x))dx   f ( x)dx   g ( x)dx
____________________________________
NOTE: Same Interval
(1). Shows the method to work Definite Integrals – like Σ
(2). IMPORTANT: Finding Area between curves
c
F). If c is between a and b , then:

b
a
c
b
a
c
f ( x)dx   f ( x)dx   f ( x )dx
_____________________________________
Placement of c important: upper bound of 1st , lower bound of 2nd .
REM: The Definite Integral is a number, so may solve the above like an equation

b

c
c
a
f ( x )dx 
f ( x )dx 
Show all the property steps to integrate.

3
1
(2 x 2  3x  5)dx
.

Given:
5
0

5
0

f ( x )dx  10
7
5

g ( x )dx  4
5
3
f ( x )dx  3
g ( x )dx  2
Show all the property steps
1)
2)
3)
4)
5)
6)
0

5

7

7

3

3
f ( x )dx 
f ( x )dx 
0
5
3
0
4 f ( x )dx 
g ( x )dx 
g ( x )dx 
  g ( x)  f ( x)  dx 
5
0
G) If f (min) is the minimum value of f (x)
and f (max) is the maximum value of f (x)
on the closed interval [a,b], then
b
f (min)(b  a)   f ( x)dx  f (max)(b  a)
a
c
Example:

Show that the integral

2
0

Show that the value of
sin( x 2 )dx cannot possibly equal 2.
1
0
x  8dx lies between 2 and 3.
Remember the Mean Value Theorem for Derivatives
F (c )  f (c ) 
And the Fundamental Theorem of Calculus

b
a
f ( x)dx
F (b)  F ( a )
ba

F (b)  F (a )
THEN
AVERAGE VALUE THEOREM (for Integrals):
f (c) 

b
a
b
 1 

  f ( x )dx
baa
f ( x )dx
ba
or
f (c) is the average of the function under consideration
i.e. On the velocity graph f (c) is the average velocity (value).
c is where that average occurs.
NOTICE: f (c) is the height of a rectangle with
the exact area of the region under the curve.
f (c)  b  a    f ( x )dx
b
a
Example:
Find the average value of the function f ( x)  x 2  2 x  1 on [ 2,4].
A car accelerates for three seconds. Its velocity in meters per second is modeled by v(t )  3t 2  2t
on t  [ 1, 4]. Find the average velocity.
Assignment 4029
PROPERTIES OF THE DEFINITE INTEGRAL
I. INTEGRATION INVOLVING THE SAME FUNCTION, f ( x )
=========================================================================
(A). Given

1
0
f ( x)dx  6

2
0

f ( x)dx  4
5
2
f ( x)dx  1
Find each of the following. Show the properties.

5
(1).

5
(3).

0
(5).
0
f ( x)dx

2
(2).
f ( x) dx

2
(4).
f ( x )dx

1
(6).
1
2
f ( x )dx
1
f ( x )dx
2
f ( x)dx
5
(B). Given

4
1
f ( x)dx  5

4
3

f ( x)dx  7
8
1
f ( x)dx  11
Find each of the following. Show the properties

8
(7).

3
(9).
(11).
4
1

4
8

3
f ( x ) dx
(8).
f ( x ) dx
f ( x) dx

8
(10).
f ( x)dx

4
(12).
4
3
4
f ( x )dx
f ( x ) dx
(C). Given

1
1

f ( x)dx  0
1
f ( x)dx  5
0
Find each of the following. Show the properties

(13).
(15).
0

1
1

f ( x)dx
(14).
3 f ( x) dx
(16).
1
1
f ( x)dx  

0
1
0
0
1
f ( x)dx
f ( x ) dx
==============================================================================
====================================================================
II. INTEGRATION INVOLVING THE SAME INTERVAL, [a , b]
=========================================================================
(C). Given

6
2
f ( x)dx  10

6
2
g ( x)dx  2
Find each of the following. Show the properties.
  f ( x)  g ( x) dx
6
(17)
(19)
2

6
2
2 f ( x) dx
  g ( x) 
6
(18)
(20)
2

2
6
f ( x)  dx
3g ( x)dx