4010-Properties of the Definite Integral (5.3)
BC Calculus
Properties of Definite Integrals
•Think rectangles
•Distance
A)

a

b

b
a
C)

b
a
f (x)
1dx  (b  a)
a
D)
D  r t
f ( x)dx  0
a
B)
A  bh
a
dx
a
f ( x)dx   f ( x)dx
b
b
kf ( x)dx  k  f ( x)dx
a
b
Properties of Definite Integrals
•Think rectangles A  b  h
•Distance
D  r t
a
E)

b
a
c
b
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.
Properties of Definite Integrals
•Think rectangles A  b  h
•Distance
D  r t
c
F) If

b
a
is between
a and b , then:
a
c
c
b
a
c
b
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
a
c
b
f ( x)dx   f ( x)dx   f ( x)dx
a
c
Examples:
Show all the steps to integrate.
3
2

1
(2 x  3x  5)dx
Examples:
GIVEN:

5
0

5
0
0
f ( x)dx  10
g ( x)dx  4
 f ( x)dx 
2)
 f ( x)dx 
3)  4 f ( x)dx 
1)
5
7
0
7
5

7
5

5
3
f ( x)dx  3
g ( x)dx  2
Examples: (cont.)
GIVEN:

5
0

5
0
4)

3

3
3
5)
0
f ( x)dx  10
g ( x)dx  4
g ( x)dx 
g ( x)dx 

7
5

5
3
f ( x)dx  3
g ( x)dx  2
Properties of Definite Integrals
* Think
rectangles
Distance
A  bh
D  r t
a
c
b
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
c
f (min)(b  a)   f ( x)dx  f (max)(b  a)
a
Example:
Show that the integral cannot possibly equal 2.
1
 sin( x )dx
2
0
Show that the value of

1
0
x  8dx
lies between 2 and 3
AVERAGE VALUE THEOREM (for Integrals)
Remember the Mean Value Theorem for Derivatives.
F (b)  F ( a )
F (c )  f (c) 
ba
And the Fundamental Theorem of Calculus
b

a
f ( x)dx

F (b)  F (a)
Then:

f (c ) 
b
a
f ( x ) dx
ba
 1 

  f ( x)dx
baa
b
AVERAGE VALUE THEOREM (for Integrals)

f (c ) 
b
a
f ( x ) dx
ba
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.
f (c)
AVERAGE VALUE THEOREM (for Integrals)

f (c ) 
b
f ( x ) dx
a
ba
f (c) is the average of the
function under consideration
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
f (c)
Method:
Find the average value of the function
on [ 2,4].
f ( x)  x  2 x  1
2
Example 2:
A car accelerates for three seconds. Its velocity in meters
per second is modeled by
t = [ 1, 4].
Find the average velocity.
v(t )  3t  2t
2
on
Last Update:
• 01/27/11
• Assignment: Worksheet
Example 3 (AP):
At different altitudes in the earth’s atmosphere, sound travels at different
speeds The speed of sound s(x) (in meters per second) can be modeled
by:
0  x  11.5
 4 x  341,
 295
11.5  x  22

 3
x  278.5
22  x  32


s ( x)   4
 3 x  254.5
32  x  50
 2

  3 x  404.5
50  x  80

 2
Where x is the altitude in kilometers. Find the average speed of sound
over the interval [ 0,80 ].
SHOW ALL PROPERTY STEPS .