lesson 24: the definite integral (section 4 version)
DESCRIPTION
The limit of Riemann Sums has a name: the definite integral. We compute a few "easy" ones and show general properties.TRANSCRIPT
![Page 1: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/1.jpg)
. . . . . .
Section5.2TheDefiniteIntegral
V63.0121, CalculusI
April16, 2009
Announcements
I MyofficeisnowWWH 624I FinalExamFriday, May8, 2:00–3:50pm
![Page 2: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/2.jpg)
. . . . . .
Outline
Recall
Thedefiniteintegralasalimit
EstimatingtheDefiniteIntegral
Propertiesoftheintegral
ComparisonPropertiesoftheIntegral
![Page 3: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/3.jpg)
. . . . . .
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
![Page 4: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/4.jpg)
. . . . . .
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
![Page 5: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/5.jpg)
. . . . . .
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
![Page 6: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/6.jpg)
. . . . . .
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
![Page 7: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/7.jpg)
. . . . . .
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
![Page 8: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/8.jpg)
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x
![Page 9: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/9.jpg)
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x
![Page 10: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/10.jpg)
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x. . .
![Page 11: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/11.jpg)
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x. . . .
![Page 12: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/12.jpg)
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x. . . . .
![Page 13: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/13.jpg)
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x. . . . . .
![Page 14: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/14.jpg)
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x. . . . . . .
![Page 15: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/15.jpg)
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x. . . . . . . .
![Page 16: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/16.jpg)
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x. . . . . . . . .
![Page 17: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/17.jpg)
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x. . . . . . . . . .
![Page 18: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/18.jpg)
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x. . . . . . . . . . .
![Page 19: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/19.jpg)
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x. . . . . . . . . . . .
![Page 20: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/20.jpg)
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x. . . . . . . . . . . . .
![Page 21: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/21.jpg)
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x. . . . . . . . . . . . . .
![Page 22: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/22.jpg)
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x. . . . . . . . . . . . . . .
![Page 23: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/23.jpg)
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x. . . . . . . . . . . . . . . .
![Page 24: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/24.jpg)
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x. . . . . . . . . . . . . . . . .
![Page 25: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/25.jpg)
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x. . . . . . . . . . . . . . . . . .
![Page 26: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/26.jpg)
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x. . . . . . . . . . . . . . . . . . .
![Page 27: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/27.jpg)
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x. . . . . . . . . . . . . . . . . . . .
![Page 28: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/28.jpg)
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x. . . . . . . . . . . . . . . . . . . . .
![Page 29: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/29.jpg)
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x......................
![Page 30: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/30.jpg)
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x.......................
![Page 31: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/31.jpg)
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x........................
![Page 32: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/32.jpg)
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x.........................
![Page 33: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/33.jpg)
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x..........................
![Page 34: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/34.jpg)
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x...........................
![Page 35: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/35.jpg)
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x............................
![Page 36: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/36.jpg)
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x.............................
![Page 37: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/37.jpg)
. . . . . .
Theoremofthe(previous)Day
TheoremIf f isacontinuousfunctionon[a,b] orhasfinitelymanyjumpdiscontinuities, then
limn→∞
Sn = limn→∞
{n∑
i=1
f(ci)∆x
}
existsandisthesamevaluenomatterwhatchoiceof ci wemade.
. .x..............................
![Page 38: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/38.jpg)
. . . . . .
Outline
Recall
Thedefiniteintegralasalimit
EstimatingtheDefiniteIntegral
Propertiesoftheintegral
ComparisonPropertiesoftheIntegral
![Page 39: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/39.jpg)
. . . . . .
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 40: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/40.jpg)
. . . . . .
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
![Page 41: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/41.jpg)
. . . . . .
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
![Page 42: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/42.jpg)
. . . . . .
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
![Page 43: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/43.jpg)
. . . . . .
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
![Page 44: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/44.jpg)
. . . . . .
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
![Page 45: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/45.jpg)
. . . . . .
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
![Page 46: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/46.jpg)
. . . . . .
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 47: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/47.jpg)
. . . . . .
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 48: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/48.jpg)
. . . . . .
Outline
Recall
Thedefiniteintegralasalimit
EstimatingtheDefiniteIntegral
Propertiesoftheintegral
ComparisonPropertiesoftheIntegral
![Page 49: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/49.jpg)
. . . . . .
EstimatingtheDefiniteIntegral
Givenapartitionof [a,b] into n pieces, let x̄i bethemidpointof[xi−1, xi]. Define
Mn =n∑
i=1
f(x̄i)∆x.
![Page 50: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/50.jpg)
. . . . . .
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 51: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/51.jpg)
. . . . . .
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 52: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/52.jpg)
. . . . . .
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 53: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/53.jpg)
. . . . . .
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 54: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/54.jpg)
. . . . . .
Outline
Recall
Thedefiniteintegralasalimit
EstimatingtheDefiniteIntegral
Propertiesoftheintegral
ComparisonPropertiesoftheIntegral
![Page 55: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/55.jpg)
. . . . . .
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 56: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/56.jpg)
. . . . . .
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 57: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/57.jpg)
. . . . . .
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 58: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/58.jpg)
. . . . . .
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 59: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/59.jpg)
. . . . . .
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 60: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/60.jpg)
. . . . . .
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 61: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/61.jpg)
. . . . . .
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 62: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/62.jpg)
. . . . . .
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.
![Page 63: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/63.jpg)
. . . . . .
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 64: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/64.jpg)
. . . . . .
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 65: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/65.jpg)
. . . . . .
Outline
Recall
Thedefiniteintegralasalimit
EstimatingtheDefiniteIntegral
Propertiesoftheintegral
ComparisonPropertiesoftheIntegral
![Page 66: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/66.jpg)
. . . . . .
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 67: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/67.jpg)
. . . . . .
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 68: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/68.jpg)
. . . . . .
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 69: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/69.jpg)
. . . . . .
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 70: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/70.jpg)
. . . . . .
Example
Estimate∫ 2
1
1xdx usingthecomparisonproperties.
SolutionSince
12≤ x ≤ 1
1forall x in [1,2], wehave
12· 1 ≤
∫ 2
1
1xdx ≤ 1 · 1
![Page 71: Lesson 24: The Definite Integral (Section 4 version)](https://reader033.vdocument.in/reader033/viewer/2022051412/549ec38bb379597b4b8b46a8/html5/thumbnails/71.jpg)
. . . . . .
Example
Estimate∫ 2
1
1xdx usingthecomparisonproperties.
SolutionSince
12≤ x ≤ 1
1forall x in [1,2], wehave
12· 1 ≤
∫ 2
1
1xdx ≤ 1 · 1