Page 1
. . . . . .
Section5.2TheDefiniteIntegral
Math1aIntroductiontoCalculus
April14, 2008
Announcements
◮ Midtermis58.3%finished◮ ProblemSessionsSunday, Thursday, 7pm, SC 310◮ OfficehoursTues 1–3pm, Weds, 2–4pmSC 323
..Image: FlickruserPhotointerference
Page 2
. . . . . .
Announcements
◮ Midtermis58.3%finished◮ ProblemSessionsSunday, Thursday, 7pm, SC 310◮ OfficehoursTues 1–3pm, Weds, 2–4pmSC 323
Page 3
. . . . . .
Outline
Lasttime: Area
Thedefiniteintegralasalimit
EstimatingtheDefiniteIntegral
Propertiesoftheintegral
ComparisonPropertiesoftheIntegral
Page 4
. . . . . .
Meetthemathematician: Archimedes
◮ 287BC –212BC (afterEuclid)
◮ Geometer◮ Weaponsengineer
Page 5
. . . . . .
Cavalieri
◮ Italian,1598–1647
◮ Revisitedtheareaproblemwithadifferentperspective
Page 6
. . . . . .
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 nth
stepbetween a and b. So
..a .b. . . . . . ..x0 .x1 .x2 .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
Page 7
. . . . . .
FormingRiemannsums
Wehavemanychoicesofhowtoapproximatethearea:
Ln = f(x0)∆x + f(x1)∆x + · · · + f(xn−1)∆x
Rn = f(x1)∆x + f(x2)∆x + · · · + f(xn)∆x
Mn = f(x0 + x1
2
)∆x + f
(x1 + x2
2
)∆x + · · · + f
(xn−1 + xn
2
)∆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
Page 8
. . . . . .
FormingRiemannsums
Wehavemanychoicesofhowtoapproximatethearea:
Ln = f(x0)∆x + f(x1)∆x + · · · + f(xn−1)∆x
Rn = f(x1)∆x + f(x2)∆x + · · · + f(xn)∆x
Mn = f(x0 + x1
2
)∆x + f
(x1 + x2
2
)∆x + · · · + f
(xn−1 + xn
2
)∆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
Page 9
. . . . . .
TheoremoftheDay
TheoremIf f isacontinuousfunctionon [a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{f(c1)∆x + f(c2)∆x + · · · + f(cn)∆x}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
Page 10
. . . . . .
Outline
Lasttime: Area
Thedefiniteintegralasalimit
EstimatingtheDefiniteIntegral
Propertiesoftheintegral
ComparisonPropertiesoftheIntegral
Page 11
. . . . . .
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
Page 12
. . . . . .
Notation/Terminology
∫ b
af(x)dx
◮∫
— integralsign (swoopy S)
◮ f(x) — integrand◮ a and b — limitsofintegration (a isthe lowerlimit and bthe upperlimit)
◮ dx —??? (aparenthesis? aninfinitesimal? avariable?)◮ Theprocessofcomputinganintegraliscalled integration
Page 13
. . . . . .
Notation/Terminology
∫ b
af(x)dx
◮∫
— integralsign (swoopy S)
◮ f(x) — integrand◮ a and b — limitsofintegration (a isthe lowerlimit and bthe upperlimit)
◮ dx —??? (aparenthesis? aninfinitesimal? avariable?)◮ Theprocessofcomputinganintegraliscalled integration
Page 14
. . . . . .
Notation/Terminology
∫ b
af(x)dx
◮∫
— integralsign (swoopy S)
◮ f(x) — integrand
◮ a and b — limitsofintegration (a isthe lowerlimit and bthe upperlimit)
◮ dx —??? (aparenthesis? aninfinitesimal? avariable?)◮ Theprocessofcomputinganintegraliscalled integration
Page 15
. . . . . .
Notation/Terminology
∫ b
af(x)dx
◮∫
— integralsign (swoopy S)
◮ f(x) — integrand◮ a and b — limitsofintegration (a isthe lowerlimit and bthe upperlimit)
◮ dx —??? (aparenthesis? aninfinitesimal? avariable?)◮ Theprocessofcomputinganintegraliscalled integration
Page 16
. . . . . .
Notation/Terminology
∫ b
af(x)dx
◮∫
— integralsign (swoopy S)
◮ f(x) — integrand◮ a and b — limitsofintegration (a isthe lowerlimit and bthe upperlimit)
◮ dx —??? (aparenthesis? aninfinitesimal? avariable?)
◮ Theprocessofcomputinganintegraliscalled integration
Page 17
. . . . . .
Notation/Terminology
∫ b
af(x)dx
◮∫
— integralsign (swoopy S)
◮ f(x) — integrand◮ a and b — limitsofintegration (a isthe lowerlimit and bthe upperlimit)
◮ dx —??? (aparenthesis? aninfinitesimal? avariable?)◮ Theprocessofcomputinganintegraliscalled integration
Page 18
. . . . . .
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
Page 19
. . . . . .
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
Page 20
. . . . . .
Outline
Lasttime: Area
Thedefiniteintegralasalimit
EstimatingtheDefiniteIntegral
Propertiesoftheintegral
ComparisonPropertiesoftheIntegral
Page 21
. . . . . .
EstimatingtheDefiniteIntegral
Givenapartitionof [a,b] into n pieces, let x̄i bethemidpointof[xi−1, xi]. Define
Mn =n∑
i=1
f(x̄i)∆x.
Page 22
. . . . . .
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
Page 23
. . . . . .
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
Page 24
. . . . . .
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
Page 25
. . . . . .
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
Page 26
. . . . . .
Outline
Lasttime: Area
Thedefiniteintegralasalimit
EstimatingtheDefiniteIntegral
Propertiesoftheintegral
ComparisonPropertiesoftheIntegral
Page 27
. . . . . .
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.
Page 28
. . . . . .
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.
Page 29
. . . . . .
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.
Page 30
. . . . . .
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.
Page 31
. . . . . .
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.
Page 32
. . . . . .
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.
Page 33
. . . . . .
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.
Page 34
. . . . . .
ExampleSuppose f and g arefunctionswith
◮∫ 4
0f(x)dx = 4
◮∫ 5
0f(x)dx = 7
◮∫ 5
0g(x)dx = 3.
Find
(a)∫ 5
0[2f(x) − g(x)] dx
(b)∫ 5
4f(x)dx.
Page 35
. . . . . .
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
Page 36
. . . . . .
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
Page 37
. . . . . .
Outline
Lasttime: Area
Thedefiniteintegralasalimit
EstimatingtheDefiniteIntegral
Propertiesoftheintegral
ComparisonPropertiesoftheIntegral
Page 38
. . . . . .
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)
Page 39
. . . . . .
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)
Page 40
. . . . . .
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)
Page 41
. . . . . .
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)
Page 42
. . . . . .
Example
Estimate∫ 2
1
1xdx usingthecomparisonproperties.
SolutionSince
12≤ x ≤ 1
1forall x in [1,2], wehave
12· 1 ≤
∫ 2
1
1xdx ≤ 1 · 1
Page 43
. . . . . .
Example
Estimate∫ 2
1
1xdx usingthecomparisonproperties.
SolutionSince
12≤ x ≤ 1
1forall x in [1,2], wehave
12· 1 ≤
∫ 2
1
1xdx ≤ 1 · 1