Post on 24-Dec-2015
MAT 1221Survey of Calculus
Section 6.4
Area and the Fundamental Theorem of Calculus
http://myhome.spu.edu/lauw
Quiz
8 minutes
Major Themes in Calculus
Abstract World
The Tangent Problem
h
afhafh
)()(lim
0
( )y f x
x a
Real World
The Velocity Problem2t
( )y f t
t a
h
afhafh
)()(lim
0
Major Themes in Calculus
Abstract World
The Tangent Problem
h
afhafh
)()(lim
0
( )y f x
x a
We do not like to use the definition
Develop techniques to deal with different functions
Major Themes in Calculus
The Area Problem
( )
( ) 0 on [ , ]
y f x
f x a b
Abstract World
1
lim ( )n
ini
A f x x
The Energy Problem
( )y f x
( )f x
Real World
Major Themes in Calculus
We do not like to use the definition
Develop techniques to deal with different functions
1
lim ( )n
ini
A f x x
The Area Problem
( )
( ) 0 on [ , ]
y f x
f x a b
Abstract World
Preview
Look at the definition of the definite integral on
Look at its relationship with the area between the graph and the -axis on
Properties of Definite Integrals The Substitution Rule for Definite
Integrals
Key
Pay attention to the overall ideas Pay less attention to the details – We are
going to use a formula to compute the definite integrals, not limits.
Example 0
]5,1[on )( 2xxf
Example 0 ]5,1[on )( 2xxf
)1(f
)5.1(f
)4(f
)5.4(f
)2(f
Use left hand end points to get an estimation
Example 0 ]5,1[on )( 2xxf
)5.2(f
)5.1(f
)5(f
)5.4(f
)2(f
Use right hand end points to get an estimation
Example 0 Observation:
What happen to the estimation if we increase the number of subintervals?
In General
ith subinterval
ix
sample point
)( ixf
In General
Suppose is a continuous function defined on , we divide the interval into n subintervals of equal width
nabx /)(
The area of the rectangle is
xxf i )(
In General
subinterval sample point
xxf i )(
In General
Sum of the area of the rectangles is
n
ii
n
xxf
xxfxxfxxfxxf
1
321
)(
)()()()(
Riemann Sum
n
ii
n
xxf
xxfxxfxxfxxf
1
321
)(
)()()()(
In General
Sum of the area of the rectangles is
Sigma Notation for summation
n
ii
n
xxf
xxfxxfxxfxxf
1
321
)(
)()()()(
In General
Sum of the area of the rectangles is
IndexInitial value (lower limit)
Final value (upper limit)
In General
Sum of the area of the rectangles is
As we increase , we get better and better estimations.
n
ii
n
xxf
xxfxxfxxfxxf
1
321
)(
)()()()(
Definition
The Definite Integral of from to
n
ii
n
b
axxfdxxf
1
)(lim)(
Definition
n
ii
n
b
axxfdxxf
1
)(lim)(
upper limit
lower limit
integrand
The Definite Integral of from to
Definition
n
ii
n
b
axxfdxxf
1
)(lim)(
Integration : Process of computing integrals
The Definite Integral of from to
Remarks
We are not going to use this limit definition to compute definite integrals.
We are going to use antiderivative (indefinite integral) to compute definite integrals.
Area and Indefinite Integrals
If on , then
from to . under"" Area )( fdxxf
b
a
b
adxxf )(
Area and Indefinite Integrals
Otherwise, the definite integral may not have obvious geometric meaning.
b
adxxf )(
Example 1
Compute by interpreting it in terms of area.
2
1)1( dxx
21
1xy1
2
1( 1)x dx
Example 1
We are going to use this example to verify our next formula.
21
1xy1
2
1( 1)x dx
Fundamental Theorem of Calculus
Suppose is continuous on and
is any antiderivative of . Then
( ) ( ) ( )b
af x dx F b F a
Remarks
To simplify the computations, we always use the antiderivative with C=0.
( ) ( ) ( )b
af x dx F b F a
Remarks
To simplify the computations, we always use the antiderivative with C=0.
We will use the following notation to stand for F(b)-F(a):
( ) ( ) ( )b
aF x F b F a
FTC
( ) ( )b b
aaf x dx F x
Suppose is continuous on and
is any antiderivative of . Then
Example 2
2
1)1( dxx
21
1xy
1
bab
axFdxxf )()(
Example 3
bab
axFdxxf )()(
2
21
2dx
x
Example 4
12 3
0
(6 8 )x x dx
bab
axFdxxf )()(
The Substitution Rule for Definite Integrals
For complicated integrands, we use a version of the substitution rule.
The Substitution Rule for Definite Integrals
The procedures for indefinite and definite integrals are similar but different.
We need to change the upper and lower limits when using a substitution.
Do not change back to the original variable.
The Substitution Rule for Definite Integrals
)(
)()()())((
bg
ag
b
aduufdxxgxgf
The Substitution Rule for Definite Integrals
)(
)()()())((
bg
ag
b
aduufdxxgxgf
Let ( ).
, ( )
, ( )
u g x
x a u g a
x b u g b
xfor range ufor range ingcorrespond
Example 51
2 4
0
10 ( 3)x x dx2Let 3
2
2
limits:
1
0
u x
dux
dxdu xdx
x u
x u
781
Example 6
22
1
1x x dx
Physical Meanings of Definite Integrals
We will not have time to discuss the exact physical meanings.
Basic Idea: The definite integral of rate of change is the net change.
Example 7 (HW 18)
A company purchases a new machine for which the rate of depreciation can be modeled by the equation below, where is the value of the machine after years.
Find the total loss of value of the machine over the first 4 years.
17000 6 , 0 5dV
t tdt