1ls3 Lectures Week11-1
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Transcript of 1ls3 Lectures Week11-1
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Announcements
Topics:
Sec1ons6.3(definiteintegral+area)and6.4(FTC)*Readthesesec1onsandstudysolvedexamplesinyour
textbook!
WorkOn:
Assignments9,20,andpartof2 Exercisesinthetextbookasassignedinthecoursepack
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Area
Howdowecalculatethe
areaofsomeirregular
shape?
Forexample,howdowe
calculatetheareaunder
thegraphoffon[a,b]?
Y
A B
F
T
Area = ?
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Area
Approach:
T
Y
A B
F
x0=
x1
x2
x3
= x4
numberof
rectangles:
widthofeach
rectangle:
Weapproximatetheareausingrectangles.
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Area
Le#handes+mate:
Area f(x0
)x + f(x1
)x + f(x2
)x + f(x3
)x
(f(x0) + f(x1) + f(x2)+ f(x3))x
f(x i)x
i= 0
3
RiemannSum
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Area
Righthandes+mate:Lettheheightofeachrectanglebegivenbythevalue
ofthefunc1onattherightendpointoftheinterval.
T
Y
A B
F
x0= x
1x2
x3 = x4
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Area
Righthandes+mate:
Area f(x1
)x + f(x2
)x + f(x3
)x + f(x4
)x
(f(x1) + f(x2)+ f(x3) + f(x4 ))x
f(x i)x
i=1
4
RiemannSum
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Area
Midpointes+mate:Lettheheightofeachrectanglebegivenbythevalue
ofthefunc1onatthemidpointoftheinterval.
T
Y
A B
F
x1
x2
x3
x4
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Area
Midpointes+mate:
Area f(x1
*)x + f(x
2
*)x + f(x
3
*)x + f(x
4
*)x
f(x i*)x
i=1
4
RiemannSum
(f(x1*) + f(x2
*) + f(x3
*) + f(x4
*))x
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RiemannSumsandthe
DefiniteIntegralDefini+on:
Thedefiniteintegralofafunc1ononthe
intervalfromatobisdefinedasalimitoftheRiemannsum
whereissomesamplepointintheinterval
and
f(x)dx = limn
f(x i*)x
i=1
n
a
b
f
x =b a
n.
xi
*
[xi1,
xi]
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Es1ma1ngUsingLeYEndpoints
-1 0 1 2 3 4 5 6 7 8
2.5
5
.
(0.5t+ 2)dt
0
5
L5 =
f(0) 1+ f(1) 1+ f(2) 1+ f(3) 1+ f(4) 1
= 2+ 2.5+ 3+ 3.5+ 4
=15
t=5 0
5=1
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Es1ma1ngUsingRightEndpoints
-1 0 1 2 3 4 5 6 7 8
2.5
5
.
(0.5t+ 2)dt
0
5
R5 = f(1) 1+ f(2) 1+ f(3) 1+ f(4) 1+ f(5) 1
= 2.5+ 3+ 3.5+ 4 + 4.5
=17.5
t=5 0
5=1
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Es1ma1ngUsingMidpoints
-1 0 1 2 3 4 5 6 7 8
2.5
5
.
(0.5t+ 2)dt
0
5
M5 = f(0.5) 1+ f(1.5) 1+ f(2.5) 1+ f(3.5) 1+ f(4.5) 1
= 2.25+ 2.75+ 3.25+ 3.75+ 4.25
=
16.25
t=5 0
5=1
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et2
dt
0
1
-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 1.25
0.25
0.5
0.75
1
.
Es1ma1ngUsingLeYEndpoints
L4 = (f(0)+ f(0.25) + f(0.5) + f(0.75)) 0.25
= (e0
2
+ e0.25
2
+ e0.5
2
+ e0.75
2
) 0.25
0.822
t=1 0
4= 0.25
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et2
dt
0
1
-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 1.25
0.25
0.5
0.75
1
.
Es1ma1ngUsingRightEndpoints
R4= (f(0.25)+ f(0.5)+ f(0.75)+ f(1)) 0.25
= (e0.25
2
+ e0.5
2
+ e0.75
2
+ e1
2
) 0.25
0.664
t=1 0
4= 0.25
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et2
dt
0
1
-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 1.25
0.25
0.5
0.75
1
.
Es1ma1ngUsingMidpoints
M4 = (f(0.125)+ f(0.375)+ f(0.625)+ f(0.875)) 0.25
= (e0.125
2
+ e0.375
2
+ e0.625
2
+ e0.875
2
) 0.25
0.749
t=1 0
4= 0.25
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TypesofIntegrals
IndefiniteIntegral
DefiniteIntegral
f(x)dx=
F(x)+
C an1deriva1veoff
func1onofx
f(x)dx = net areaa
b
number
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TheDefiniteIntegral
Interpreta+on:
If,thenthedefinite
integralistheareaunderthecurvefrom
atob.
Y
A B
F
T
f 0
Area = f(x)dxa
b
y = f(x)
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TheDefiniteIntegral
Interpreta+on:
Ifisbothposi1veand
nega1ve,thenthedefiniteintegralrepresentsthe
NETorSIGNEDarea,i.e.
theareaabovethexaxis
andbelowthegraphoffminustheareabelowthe
xaxisandabovef
f
f(x)dx1
4
= net area
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DefiniteIntegralsandArea
Example:
Evaluatethefollowingintegralsbyinterpre1ng
eachintermsofarea.
(a) (b)
(c)
(x 1) dx0
3
1 x 2 dx0
1
sinx dx
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Proper1esofIntegrals
Assumethatf(x)andg(x)arecon1nuous
func1onsanda,b,andcarerealnumberssuch
thata
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Proper1esofIntegrals
Assumethatf(x)andg(x)arecon1nuous
func1onsanda,b,andcarerealnumberssuch
thata
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Summa1onPropertyofthe
DefiniteIntegral(6)Supposef(x)iscon1nuousontheinterval
fromatobandthat
Then
a c b.
f(x)dxa
b
= f(x)dxa
c
+ f(x)dxc
b
.
Y
A B
F
T
C
f(x)dxa
c
f(x)dxc
b
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TypesofIntegrals
IndefiniteIntegral
DefiniteIntegral
f(x)dx=
F(x)+
C an1deriva1veoff
func1onofx
f(x)dx = net areaa
b
number
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TheFundamentalTheoremofCalculus
Ifiscon5nuousonthen
whereisanyan1deriva1veof,i.e.,
f(x)dxa
b
= F(x) ab = F(b) F(a)
f
[a, b],
F
f
F'=
f.
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Differen1a1onandIntegra1on
asInverseProcessesIffisintegratedandthendifferen1ated,we
arrivebackattheoriginalfunc1onf.
IfFisdifferen1atedandthenintegrated,wearrivebackattheoriginalfunc1onF.
d
dxF(x)dx = F(x)
a
b
a
b
d
dxf(t)dt
a
x
= f(x)
FTCII
FTCI
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Applica1on
Example:
Supposethatthegrowthrateofafishisgivenby
thedifferen1alequa1on
wheretismeasuredinyearsandLismeasuredincen1metresandthefishwas0.0cmatage
t=0(1memeasuredfromfer1liza1on).
dL
dt= 6.48e
0.09t
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