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MAT 1221 Survey of Calculus. Section 6.4 Area and the Fundamental Theorem of Calculus. http://myhome.spu.edu/lauw. Quiz. 8 minutes. Bonus Event (5/9). Your feedback is very helpful to the speaker. Bonus Event. Friday 5/23; 5:10- 5:40 (Hedging), 5:45-6:15 (Genetics Inbreeding Problem) - PowerPoint PPT Presentation
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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
hafhaf
h
)()(lim0
( )y f xx a
Real World
The Velocity Problem2t
( )y f tt a
hafhaf
h
)()(lim0
Major Themes in Calculus
Abstract World
The Tangent Problem
hafhaf
h
)()(lim0
( )y f xx a
We do not like to use the definition
Develop techniques to deal with different functions
Major Themes in CalculusThe Area Problem
( )( ) 0 on [ , ]
y f xf x a b
Abstract World
1
lim ( )n
in i
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
in i
A f x x
The Area Problem
( )( ) 0 on [ , ]
y f xf 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 GeneralSuppose 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 GeneralSum of the area of the rectangles is
n
ii
n
xxf
xxfxxfxxfxxf
1
321
)(
)()()()(
Riemann Sum
n
ii
n
xxf
xxfxxfxxfxxf
1
321
)(
)()()()(
In GeneralSum of the area of the rectangles is
Sigma Notation for summation
n
ii
n
xxf
xxfxxfxxfxxf
1
321
)(
)()()()(
In GeneralSum of the area of the rectangles is
IndexInitial value (lower limit)
Final value (upper limit)
In GeneralSum of the area of the rectangles is
As we increase , we get better and better estimations.
n
ii
n
xxf
xxfxxfxxfxxf
1
321
)(
)()()()(
DefinitionThe Definite Integral of from to
n
iin
b
axxfdxxf
1
)(lim)(
Definition
n
iin
b
axxfdxxf
1
)(lim)(
upper limit
lower limit
integrand
The Definite Integral of from to
Definition
n
iin
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 IntegralsIf on , then
from to . under"" Area )( fdxxf
b
a
b
adxxf )(
Area and Indefinite IntegralsOtherwise, the definite integral may not have obvious geometric meaning.
b
adxxf )(
Example 1Compute by interpreting it in terms of area.
2
1)1( dxx
21
1xy1
2
1( 1)x dx
Example 1We are going to use this example to verify our next formula.
21
1xy1
2
1( 1)x dx
Fundamental Theorem of CalculusSuppose is continuous on andis 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
2 dxx
Example 41
2 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 xx a u g ax b u g b
xfor range ufor range ingcorrespond
Example 51
2 4
0
10 ( 3)x x dx2Let 3
2
2limits:
10
u xdu xdxdu xdx
x ux u
781
Example 62
2
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
