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. . . . . .
Section5.2TheDefiniteIntegral
V63.0121, CalculusI
April16, 2009
Announcements
I MyofficeisnowWWH 624I FinalExamFriday, May8, 2:00–3:50pm
. . . . . .
Outline
Recall
Thedefiniteintegralasalimit
EstimatingtheDefiniteIntegral
Propertiesoftheintegral
ComparisonPropertiesoftheIntegral
. . . . . .
Cavalieri’smethodingeneralLet f beapositivefunctiondefinedontheinterval [a,b]. Wewanttofindtheareabetween x = a, x = b, y = 0, and y = f(x).Foreachpositiveinteger n, divideuptheintervalinto n pieces.
Then ∆x =b− an
. Foreach i between 1 and n, let xi bethe ith
stepbetween a and b. So
. .x..x0
..x1
..xi
..xn−1
..xn.. . . .. . .
x0 = a
x1 = x0 + ∆x = a +b− an
x2 = x1 + ∆x = a + 2 · b− an
. . .
xi = a + i · b− an
. . .
xn = a + n · b− an
= b
. . . . . .
FormingRiemannsumsWehavemanychoicesofrepresentativepointstoapproximatetheareaineachsubinterval.
leftendpoints…
Ln =n∑
i=1
f(xi−1)∆x
. .x. . . . . . .Ingeneral, choose ci tobeapointinthe ithinterval [xi−1, xi].Formthe Riemannsum
Sn = f(c1)∆x + f(c2)∆x + · · · + f(cn)∆x =n∑
i=1
f(ci)∆x
. . . . . .
FormingRiemannsumsWehavemanychoicesofrepresentativepointstoapproximatetheareaineachsubinterval.
rightendpoints…
Rn =n∑
i=1
f(xi)∆x
. .x. . . . . . .Ingeneral, choose ci tobeapointinthe ithinterval [xi−1, xi].Formthe Riemannsum
Sn = f(c1)∆x + f(c2)∆x + · · · + f(cn)∆x =n∑
i=1
f(ci)∆x
. . . . . .
FormingRiemannsumsWehavemanychoicesofrepresentativepointstoapproximatetheareaineachsubinterval.
midpoints…
Mn =n∑
i=1
f(xi−1 + xi
2
)∆x
. .x. . . . . . .Ingeneral, choose ci tobeapointinthe ithinterval [xi−1, xi].Formthe Riemannsum
Sn = f(c1)∆x + f(c2)∆x + · · · + f(cn)∆x =n∑
i=1
f(ci)∆x
. . . . . .
FormingRiemannsumsWehavemanychoicesofrepresentativepointstoapproximatetheareaineachsubinterval.
randompoints…
. .x. . . . . . .Ingeneral, choose ci tobeapointinthe ithinterval [xi−1, xi].Formthe Riemannsum
Sn = f(c1)∆x + f(c2)∆x + · · · + f(cn)∆x =n∑
i=1
f(ci)∆x
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x. . .
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x. . . .
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x. . . . .
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x. . . . . .
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x. . . . . . .
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x. . . . . . . .
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x. . . . . . . . .
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x. . . . . . . . . .
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x. . . . . . . . . . .
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x. . . . . . . . . . . .
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x. . . . . . . . . . . . .
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x. . . . . . . . . . . . . .
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x. . . . . . . . . . . . . . .
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x. . . . . . . . . . . . . . . .
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x. . . . . . . . . . . . . . . . .
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x. . . . . . . . . . . . . . . . . .
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x. . . . . . . . . . . . . . . . . . .
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x. . . . . . . . . . . . . . . . . . . .
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x. . . . . . . . . . . . . . . . . . . . .
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x......................
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x.......................
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x........................
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x.........................
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x..........................
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x...........................
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x............................
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x.............................
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x..............................
. . . . . .
Outline
Recall
Thedefiniteintegralasalimit
EstimatingtheDefiniteIntegral
Propertiesoftheintegral
ComparisonPropertiesoftheIntegral
. . . . . .
Thedefiniteintegralasalimit
DefinitionIf f isafunctiondefinedon [a,b], the definiteintegralof f from ato b isthenumber∫ b
af(x)dx = lim
∆x→0
n∑i=1
f(ci) ∆x
. . . . . .
Notation/Terminology
∫ b
af(x)dx
I∫
— integralsign (swoopy S)
I f(x) — integrandI a and b — limitsofintegration (a isthe lowerlimit and bthe upperlimit)
I dx —??? (aparenthesis? aninfinitesimal? avariable?)I Theprocessofcomputinganintegraliscalled integration or
quadrature
. . . . . .
Notation/Terminology
∫ b
af(x)dx
I∫
— integralsign (swoopy S)
I f(x) — integrandI a and b — limitsofintegration (a isthe lowerlimit and bthe upperlimit)
I dx —??? (aparenthesis? aninfinitesimal? avariable?)I Theprocessofcomputinganintegraliscalled integration or
quadrature
. . . . . .
Notation/Terminology
∫ b
af(x)dx
I∫
— integralsign (swoopy S)
I f(x) — integrand
I a and b — limitsofintegration (a isthe lowerlimit and bthe upperlimit)
I dx —??? (aparenthesis? aninfinitesimal? avariable?)I Theprocessofcomputinganintegraliscalled integration or
quadrature
. . . . . .
Notation/Terminology
∫ b
af(x)dx
I∫
— integralsign (swoopy S)
I f(x) — integrandI a and b — limitsofintegration (a isthe lowerlimit and bthe upperlimit)
I dx —??? (aparenthesis? aninfinitesimal? avariable?)I Theprocessofcomputinganintegraliscalled integration or
quadrature
. . . . . .
Notation/Terminology
∫ b
af(x)dx
I∫
— integralsign (swoopy S)
I f(x) — integrandI a and b — limitsofintegration (a isthe lowerlimit and bthe upperlimit)
I dx —??? (aparenthesis? aninfinitesimal? avariable?)
I Theprocessofcomputinganintegraliscalled integration orquadrature
. . . . . .
Notation/Terminology
∫ b
af(x)dx
I∫
— integralsign (swoopy S)
I f(x) — integrandI a and b — limitsofintegration (a isthe lowerlimit and bthe upperlimit)
I dx —??? (aparenthesis? aninfinitesimal? avariable?)I Theprocessofcomputinganintegraliscalled integration or
quadrature
. . . . . .
Thelimitcanbesimplified
TheoremIf f iscontinuouson [a,b] orif f hasonlyfinitelymanyjumpdiscontinuities, then f isintegrableon [a,b]; thatis, thedefinite
integral∫ b
af(x)dx exists.
TheoremIf f isintegrableon [a,b] then∫ b
af(x)dx = lim
n→∞
n∑i=1
f(xi)∆x,
where
∆x =b− an
and xi = a + i∆x
. . . . . .
Thelimitcanbesimplified
TheoremIf f iscontinuouson [a,b] orif f hasonlyfinitelymanyjumpdiscontinuities, then f isintegrableon [a,b]; thatis, thedefinite
integral∫ b
af(x)dx exists.
TheoremIf f isintegrableon [a,b] then∫ b
af(x)dx = lim
n→∞
n∑i=1
f(xi)∆x,
where
∆x =b− an
and xi = a + i∆x
. . . . . .
Outline
Recall
Thedefiniteintegralasalimit
EstimatingtheDefiniteIntegral
Propertiesoftheintegral
ComparisonPropertiesoftheIntegral
. . . . . .
EstimatingtheDefiniteIntegral
Givenapartitionof [a,b] into n pieces, let x̄i bethemidpointof[xi−1, xi]. Define
Mn =n∑
i=1
f(x̄i)∆x.
. . . . . .
Example
Estimate∫ 1
0
41 + x2
dx usingthemidpointruleandfourdivisions.
SolutionThepartitionis 0 <
14
<12
<34
< 1, sotheestimateis
M4 =14
(4
1 + (1/8)2+
41 + (3/8)2
+4
1 + (5/8)2+
41 + (7/8)2
)
=14
(4
65/64+
473/64
+4
89/64+
4113/64
)=
150, 166,78447, 720, 465
≈ 3.1468
. . . . . .
Example
Estimate∫ 1
0
41 + x2
dx usingthemidpointruleandfourdivisions.
SolutionThepartitionis 0 <
14
<12
<34
< 1, sotheestimateis
M4 =14
(4
1 + (1/8)2+
41 + (3/8)2
+4
1 + (5/8)2+
41 + (7/8)2
)
=14
(4
65/64+
473/64
+4
89/64+
4113/64
)=
150, 166,78447, 720, 465
≈ 3.1468
. . . . . .
Example
Estimate∫ 1
0
41 + x2
dx usingthemidpointruleandfourdivisions.
SolutionThepartitionis 0 <
14
<12
<34
< 1, sotheestimateis
M4 =14
(4
1 + (1/8)2+
41 + (3/8)2
+4
1 + (5/8)2+
41 + (7/8)2
)=
14
(4
65/64+
473/64
+4
89/64+
4113/64
)
=150, 166,78447, 720, 465
≈ 3.1468
. . . . . .
Example
Estimate∫ 1
0
41 + x2
dx usingthemidpointruleandfourdivisions.
SolutionThepartitionis 0 <
14
<12
<34
< 1, sotheestimateis
M4 =14
(4
1 + (1/8)2+
41 + (3/8)2
+4
1 + (5/8)2+
41 + (7/8)2
)=
14
(4
65/64+
473/64
+4
89/64+
4113/64
)=
150, 166,78447, 720, 465
≈ 3.1468
. . . . . .
Outline
Recall
Thedefiniteintegralasalimit
EstimatingtheDefiniteIntegral
Propertiesoftheintegral
ComparisonPropertiesoftheIntegral
. . . . . .
Propertiesoftheintegral
Theorem(AdditivePropertiesoftheIntegral)Let f and g beintegrablefunctionson [a,b] and c aconstant.Then
1.∫ b
ac dx = c(b− a)
2.∫ b
a[f(x) + g(x)] dx =
∫ b
af(x)dx +
∫ b
ag(x)dx.
3.∫ b
acf(x)dx = c
∫ b
af(x)dx.
4.∫ b
a[f(x) − g(x)] dx =
∫ b
af(x)dx−
∫ b
ag(x)dx.
. . . . . .
Propertiesoftheintegral
Theorem(AdditivePropertiesoftheIntegral)Let f and g beintegrablefunctionson [a,b] and c aconstant.Then
1.∫ b
ac dx = c(b− a)
2.∫ b
a[f(x) + g(x)] dx =
∫ b
af(x)dx +
∫ b
ag(x)dx.
3.∫ b
acf(x)dx = c
∫ b
af(x)dx.
4.∫ b
a[f(x) − g(x)] dx =
∫ b
af(x)dx−
∫ b
ag(x)dx.
. . . . . .
Propertiesoftheintegral
Theorem(AdditivePropertiesoftheIntegral)Let f and g beintegrablefunctionson [a,b] and c aconstant.Then
1.∫ b
ac dx = c(b− a)
2.∫ b
a[f(x) + g(x)] dx =
∫ b
af(x)dx +
∫ b
ag(x)dx.
3.∫ b
acf(x)dx = c
∫ b
af(x)dx.
4.∫ b
a[f(x) − g(x)] dx =
∫ b
af(x)dx−
∫ b
ag(x)dx.
. . . . . .
Propertiesoftheintegral
Theorem(AdditivePropertiesoftheIntegral)Let f and g beintegrablefunctionson [a,b] and c aconstant.Then
1.∫ b
ac dx = c(b− a)
2.∫ b
a[f(x) + g(x)] dx =
∫ b
af(x)dx +
∫ b
ag(x)dx.
3.∫ b
acf(x)dx = c
∫ b
af(x)dx.
4.∫ b
a[f(x) − g(x)] dx =
∫ b
af(x)dx−
∫ b
ag(x)dx.
. . . . . .
MorePropertiesoftheIntegral
Conventions: ∫ a
bf(x)dx = −
∫ b
af(x)dx
∫ a
af(x)dx = 0
Thisallowsustohave
5.∫ c
af(x)dx =
∫ b
af(x)dx +
∫ c
bf(x)dx forall a, b, and c.
. . . . . .
MorePropertiesoftheIntegral
Conventions: ∫ a
bf(x)dx = −
∫ b
af(x)dx
∫ a
af(x)dx = 0
Thisallowsustohave
5.∫ c
af(x)dx =
∫ b
af(x)dx +
∫ c
bf(x)dx forall a, b, and c.
. . . . . .
MorePropertiesoftheIntegral
Conventions: ∫ a
bf(x)dx = −
∫ b
af(x)dx
∫ a
af(x)dx = 0
Thisallowsustohave
5.∫ c
af(x)dx =
∫ b
af(x)dx +
∫ c
bf(x)dx forall a, b, and c.
. . . . . .
ExampleSuppose f and g arefunctionswith
I∫ 4
0f(x)dx = 4
I∫ 5
0f(x)dx = 7
I∫ 5
0g(x)dx = 3.
Find
(a)∫ 5
0[2f(x) − g(x)] dx
(b)∫ 5
4f(x)dx.
. . . . . .
SolutionWehave
(a) ∫ 5
0[2f(x) − g(x)] dx = 2
∫ 5
0f(x)dx−
∫ 5
0g(x)dx
= 2 · 7− 3 = 11
(b) ∫ 5
4f(x)dx =
∫ 5
0f(x)dx−
∫ 4
0f(x)dx
= 7− 4 = 3
. . . . . .
SolutionWehave
(a) ∫ 5
0[2f(x) − g(x)] dx = 2
∫ 5
0f(x)dx−
∫ 5
0g(x)dx
= 2 · 7− 3 = 11
(b) ∫ 5
4f(x)dx =
∫ 5
0f(x)dx−
∫ 4
0f(x)dx
= 7− 4 = 3
. . . . . .
Outline
Recall
Thedefiniteintegralasalimit
EstimatingtheDefiniteIntegral
Propertiesoftheintegral
ComparisonPropertiesoftheIntegral
. . . . . .
ComparisonPropertiesoftheIntegral
TheoremLet f and g beintegrablefunctionson [a,b].
6. If f(x) ≥ 0 forall x in [a,b], then∫ b
af(x)dx ≥ 0
7. If f(x) ≥ g(x) forall x in [a,b], then∫ b
af(x)dx ≥
∫ b
ag(x)dx
8. If m ≤ f(x) ≤ M forall x in [a,b], then
m(b− a) ≤∫ b
af(x)dx ≤ M(b− a)
. . . . . .
ComparisonPropertiesoftheIntegral
TheoremLet f and g beintegrablefunctionson [a,b].
6. If f(x) ≥ 0 forall x in [a,b], then∫ b
af(x)dx ≥ 0
7. If f(x) ≥ g(x) forall x in [a,b], then∫ b
af(x)dx ≥
∫ b
ag(x)dx
8. If m ≤ f(x) ≤ M forall x in [a,b], then
m(b− a) ≤∫ b
af(x)dx ≤ M(b− a)
. . . . . .
ComparisonPropertiesoftheIntegral
TheoremLet f and g beintegrablefunctionson [a,b].
6. If f(x) ≥ 0 forall x in [a,b], then∫ b
af(x)dx ≥ 0
7. If f(x) ≥ g(x) forall x in [a,b], then∫ b
af(x)dx ≥
∫ b
ag(x)dx
8. If m ≤ f(x) ≤ M forall x in [a,b], then
m(b− a) ≤∫ b
af(x)dx ≤ M(b− a)
. . . . . .
ComparisonPropertiesoftheIntegral
TheoremLet f and g beintegrablefunctionson [a,b].
6. If f(x) ≥ 0 forall x in [a,b], then∫ b
af(x)dx ≥ 0
7. If f(x) ≥ g(x) forall x in [a,b], then∫ b
af(x)dx ≥
∫ b
ag(x)dx
8. If m ≤ f(x) ≤ M forall x in [a,b], then
m(b− a) ≤∫ b
af(x)dx ≤ M(b− a)
. . . . . .
Example
Estimate∫ 2
1
1xdx usingthecomparisonproperties.
SolutionSince
12≤ x ≤ 1
1forall x in [1,2], wehave
12· 1 ≤
∫ 2
1
1xdx ≤ 1 · 1
. . . . . .
Example
Estimate∫ 2
1
1xdx usingthecomparisonproperties.
SolutionSince
12≤ x ≤ 1
1forall x in [1,2], wehave
12· 1 ≤
∫ 2
1
1xdx ≤ 1 · 1