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4010-Properties of the Definite Integral (5.3)
BC Calculus
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Properties of Definite Integrals
A)
B)
C)
D)
A b hD r t
• Think rectangles• Distance
( ) 0a
af x dx
1 ( )b
adx b a ( ) ( )
b a
a bf x dx f x dx
a dx b
f (x)
( ) ( )b b
a akf x dx k f x dx
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Properties of Definite Integrals
A b hD r t
• Think rectangles• Distance
a c b
( ( ) ( )) ( ) ( )b b b
a a af x g x dx f x dx g x dx E)
NOTE: Same Interval
(2). IMPORTANT: Finding Area between curves.
(1). Shows the method to work Definite Integrals – like Σ
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Properties of Definite Integrals
A b hD r t
• Think rectangles• Distance
a c b
( ) ( ) ( )b c b
a a cf x dx f x dx f x dx
F) If c is between a and b , then:
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 b
a a cf x dx f x dx f x dx
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Examples:
Show all the steps to integrate.3 2
1(2 3 5)x x dx
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Examples:
GIVEN: 5
0( ) 10f x dx
7
5( ) 3f x dx
5
0( ) 4g x dx
1)
7
0( )f x dx 2)
0
5( )f x dx
3)7
54 ( )f x dx
5
3( ) 2g x dx
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Examples: (cont.)
GIVEN: 5
0( ) 10f x dx
7
5( ) 3f x dx
5
0( ) 4g x dx
4)
5)
5
3( ) 2g x dx
3
3( )g x dx
3
0( )g x dx
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Properties of Definite Integrals
Distance
A b hD r t
* Think rectangles
a c b
(min)( ) ( ) (max)( )c
af b a f x dx f b a
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
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Example:
1 2
0sin( )x dx
Show that the integral cannot possibly equal 2.
Show that the value of lies between 2 and 3 1
08x dx
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AVERAGE VALUE THEOREM (for Integrals)Remember the Mean Value Theorem for Derivatives.
( ) ( )( ) ( ) F b F aF c f cb a
And the Fundamental Theorem of Calculus
( ) ( ) ( )b
af x dx F b F a
Then:
( )( )
b
af x dx
f cb a
1 ( )b
a
f x dxb a
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AVERAGE VALUE THEOREM (for Integrals)
( )( )
b
af x dx
f cb a
f (c)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.
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AVERAGE VALUE THEOREM (for Integrals)
( )( )
b
af x dx
f cb a
f (c)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.
( ) ( )b
af c b a f x dx
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Method:
Find the average value of the function
on [ 2,4].
2( ) 2 1f x x x
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Example 2:A car accelerates for three seconds. Its velocity in meters
per second is modeled by on
t = [ 1, 4].
Find the average velocity.
2( ) 3 2v t t t
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Last Update:
• 01/27/11
• Assignment: Worksheet
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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:
4 341, 0 11.5295 11.5 22
3 278.5 22 324( )3 254.5 32 5023 404.5 50 802
x xx
x xs x
x x
x x
Where x is the altitude in kilometers. Find the average speed of sound over the interval [ 0,80 ].
SHOW ALL PROPERTY STEPS .