The document discusses the concept of double integrals, particularly over rectangular regions, and their applications in calculating volumes. It provides definitions, properties, examples, and theorems related to double integrals and iterated integrals. Additionally, the document touches upon integration in polar coordinates and presents exercises for evaluation.

1. Multiple Integral

2. Double Integrals over Rectangles
Remark ：
)(point.sample
acalledisx1,2,...n,i],x,[xxChoose4.
Pofnorm}Thex,,x,xmax{|P|3.
n,1,2,i,x-xxDefine2.
b],[aofpartitionacalledisPThen
bxxxaand}x,,x,{xPLet1.
ii1-ii
n21
1-iii
n10n10
取樣點
=∈
∆∆∆=
==∆
=<<<==




3. ∑∫ =
→
=∆=
n
1i
ii
b
a 0P||
SofareaThex)xf(limf(x)dx
b],[aonfofintegraldefiniteThe5.
f(x)}y0b,xa|y){(x,S ≤≤≤≤=

4. [2,3][1,2]2.
4}y21,x0|Ry){(x,[2,4][0,1]1.
Rectangle:Example
d}ycb,xa|Ry){(x,d][c,b][a,R
RrectangleclosedA6.
2
2
×
≤≤≤≤∈=×
≤≤≤≤∈=×=

5. Double Integrals and Volumes
V(S)-SofvolumethefindTo
y)}f(x,zR,0y)(x,|Rz)y,{(x,Sd],[c,b][a,RLet 3
≤≤∈∈=×=
lssubintervaintoRrectangletheDivide
f(x)dxdefinetoSimilarly
b
a∫

6. ∑∑
∑∑
= =
= =
∆≈
∆
∆∆=∆
==×=
==∆
=∆=
n
1j
m
1i
ij
*
ij
*
ij
n
1j
m
1i
ij
*
ij
*
ij
ij
*
ij
*
ij
jiijij
j1-ji1-iij
1-jjjj1-j
1-iii
i1-i
A)y,f(xV(S)i.e
A)y,f(xeapproximatcanSofvolumeThe
Reachin)y,(xpointsampleachoose
yxAisRofareaThe
n1,j;m1,i]y,[y]x,[xRDefine
n1,2,j,y-yy],y,[ylsubintervaninto
dividedisd][c,and,x-xxm,1,2,i
]x,[xlsubintervamintodividedisb][a,




7. n1,2j;m1,2i,R
diagonallongesttheoflengththedenote|P|Let
ij  ==

8. ∑∑∫∫
∫∫
= =
→
∆=
m
1i
n
1j
ij
*
ij
*
ij
0P||
R
R
A)y,f(xlimy)dAf(x,
y)dAf(x,isRrectangleover thefofintegraldoubleThe
:Definition

9. ∫∫ ∫∫
∫∫ ∫∫ ∫∫
∫∫ ∫∫
≥
∈∀≥
+=+
=
R R
R R R
R R
y)dAg(x,y)dAf(x,
thenR,y)(x,y)g(x,y)f(x,If3.
y)dAg(x,y)dAf(x,y))dAg(x,y)(f(x,2.
y)dAf(x,cy)dAcf(x,1.
:properties
existslimitthisif

10. Ronintegrableisfthencurves,smooth
ofnumberfiniteaonexceptRoncontinuousisfIf(ii)
RonintegrableisfthenR,oncontinuousisfIf(i)
Rrectabgleclosedon theboundedbefLet
1Theorem
Roverfofintegraldoublethecalledisy)dAf(x,2.
existA)y,f(xlimifR,onintegrableisf1.
:Definition
R
m
1i
n
1j
ij
*
ij
*
ij
0P||
∫∫
∑∑= =
→
∆

11. ww),(o,oncontinuousnotisf
(0,1)oncontinuousnotisf
0x0,
0x,
x
y
y)f(x,3.
Roncontinuousisf
)[0,)[0,Rx,yxy)f(x,2.
Roncontinuousisf
][0,2][0,Ry)(x,sinxy,y)f(x,1.
:Example
RoncontinuousisfthenR,b)(a,allatcontinuousisfIf2.
b)(a,atcontinuousisfthenb),f(a,y)f(x,limIf1.
:Definition
2
b)(a,y)(x,
∀




=
≠
=
∞×∞=+=
×=∈=
∈
=
→
ππ

12. Iterated Integrals
integraliteratedancalledis)dxy)dyf(x,(A(x)dx
y)dxf(x,B(y)y)dy,f(x,A(x)Let1.
d][c,b][a,Ry),f(x,functionFor
:Remark
)dyydxx(A(y)dyConsider
y
3
26
1
3
3
x
yydxxA(y)Lety,Fixed
b
a
d
c
b
a
d
c
b
a
2
0
3
1
2
2
0
3
3
1
2
∫ ∫∫
∫ ∫
∫ ∫∫
∫
=
==
×=
=
=⋅==

13. ∫ ∫
∫ ∫
∫ ∫
∫ ∫∫ ∫
∫ ∫∫ ∫
∫ ∫∫ ∫
+
+
+
=
=
3
0
2
1
8
0
4
0
2
4
0
8
0
2
3
0
4
1
2
3
0
4
1
2
d
c
b
a
d
c
b
a
b
a
d
c
b
a
d
c
2y)dxdy(3x(v)
)dxdyy8x-(64
4
1
(iv)
)dydxy8x-(64
4
1
(iii)
ydydxx(ii)ydxdyx(i)
evaluate:Example
)dxy)dyf(x,(y)dydxf(x,3.
)dxy)dyf(x,(y)dydxf(x,2.

14. 1}y02,x-1|y){(x,Rwhere,dA
x1
y1
Find4.
][0,][0,Rwhere,xcosxydAFind3.
0-)
0
sin2y
2
1
(-
cos2y)dy-(1dy
0
2
(-cosxy)ysinxydxdyBut
?ysinxydydxysinxydA:Ans
][0,[0,2]Rwhere,ysinxydAFind2.
[1,3][0,2]RwheredA,)3y-(xFind1.
:Ex
y)dydxf(x,y)dxdyf(x,y)dAf(x,
thed],[c,b][a,RoncontinuousisfIf
Theorem)s(Fubini'2Theorem
R
R
000
2
0
2
0 0
R
R
R
2
b
a
d
c
d
c
b
a
R
≤≤≤≤=
+
+
×=
===
==
==
×=
×=
==
×=
∫∫
∫∫
∫∫∫ ∫
∫ ∫∫∫
∫∫
∫∫
∫ ∫∫ ∫∫∫
ππ
ππ
π
π
π
πππ
π

15. { }
1y)dAsin(xD
thatshow[0,1],[0,1]RIf2.
2y14,x34,
2y13,x11,
1y04,x12,
y)f(x,2.
2y04,x33,
2y03,x12,
y)f(x,1.
functiongiventheisfwherey)dA,f(x,
Evaluate,2y03,x1|y)(x,RLet1.
Exercises
R
R
≤+≤
×=





≤≤≤≤
≤≤<≤
≤≤≤≤
=



≤≤≤<
≤≤≤≤
=
≤≤≤≤=
∫∫
∫∫

16. ∫∫
∫∫
∫∫
∫∫
∫∫ ∫∫ ∫∫
+
+
===
≤≤≤≤=
×=≤≤≤≤=
R
R
R
R
R R R
2
1
y))dA4g(x,y)(3f(x,4.
4)dAy)(2f(x,3.
y)dA3g(x,2.
y))dAg(x,-y)(4f(x,1.
Evaluate
2y)dAg(x,6,y)dAg(x,4,y)dAf(x,thatSuppose
2}y12,x0|y){(x,R
[0,2][0,2]R1},y02,x0|y){(x,RLet3.
1
2

17. ∫ ∫
∫ ∫
∫ ∫
∫ ∫
∫ ∫
∫ ∫
∫ ∫
+
=
2
2-
1
1-
2
2-
3
1
1-
2
2
2-
1
1-
32
2
0
2
0 2
0
1
0
1
0
1
0
xy
2
0
3
1
2
dydx|y|[x](vii)
dydxy][x(vi)
dydx|yx|(v)
dydx
x1
y
(iv)
xsinydxdy(iii)
dxdyxe(ii)
yxy)f(x,wherey)dydx,f(x,(i)
integralsinteratedtheofeachEvaluate4.
π

18. { }
2}y,13x0|y){(x,R,dAx1xy(v)
[0,1][0,1]R,dAxye(iv)
1}y02,x-1|y){(x,RdA,
y2
x1
(iii)
]
3
[0,]
6
[0,Ry)dA,xcos(x(ii)
3y02,x1y)(x,R)dA,3xy-(2y(i)
integraldoubletheCalculate5.
R
2
R
yx
R
R
R
32
22
≤≤≤≤=+
×=
≤≤≤≤=
+
+
×=+
≤≤≤≤=
∫∫
∫∫
∫∫
∫∫
∫∫
ππ

19. ∫ ∫ ∫∫
∫ ∫∫ ∫
∫ ∫ ∫∫
+
≤≤≤≤
++=
+
1
0
4
0
3
2x
y
y
0
x
1
0
1
y
3
1
0
3
3y
x
1
0
1
0
x2
x-1
2
x
0
2
dydxe(vi)dxdy2ye(v)
dxdyx1(iv)dxdye(iii)
)dydxy-(3x(ii)dydxcosx(i)
integralinteratedtheEvaluate7.
4)y10,x(-1|y){(x,
rectangletheaboveand15y2xZ
planeunder thelyingsolidtheofvolumetheFind6.
2
2
2

20. 1zyxand
0z0,y0,xplanesby theBounded(ii)
2zy
and2yxcylindersby theBounded(i)
solidgiventheofvolumetheFind9.
1}yx|y){(x,DwheredA,y-x-1Evaluate8.
222
222
22
D
22
=++
===
=+
=+
≤+=∫∫

21. 

 ∈
=
DinnotbutRinisy)(x,if0
Dy)(x,ify)f(x,
y)F(x,
Double Integral over General Regions
functionnewaDefineD.on
definedfunctionaisfR,DandregionboundedabeDLet ⊂

22. function.continuoustwoareh,hwhere
d},yc(y),hx(y)h|y){(x,D
ifIItypeofbetosaidisDregionplaneA3.
function.continuoustwoareg,gwhere
(x)},gy(x)gb,xa|y){(x,D
ifItypeofbetosaidisDregionplaneA2.
y)dAF(x,y)dAf(x,
isDoverfofintegraldoubleThe1.
:Definition
21
21
21
21
D R
≤≤≤≤=
≤≤≤≤=
=∫∫ ∫∫

23. d}yc(y),hx(y)h|y){(x,Dwhere
y)dxdyf(x,y)dAf(x,
thenDregionIItypeaoncontinuousisfIf2.
y)dydxf(x,y)dAf(x,then
(x)},gy(x)gb,xa|y){(x,D
such thatDregionItypeaoncontinuousisfIf1.
:Properties
IIType},y1x2y1,y-1|y){(x,D2.
IType1},ysinx,x0|y){(x,D1.
:Example
21
D
d
c
(y)h
h
D
b
a
(x)g
g
21
22
2
1
2
1(y)
2
1(x)
≤≤≤≤=
=
=
≤≤≤≤=
+≤≤≤≤=
≤≤≤≤=
∫∫ ∫ ∫
∫∫ ∫ ∫
π

24. 2
2
1
-1
2
3
1-
1
)x
2
1
-xx
2
3
x
4
1
-x
2
1
(
dx4x-x
2
3
3x
2
3
2x-xx
)dx)(2x-)x((1
2
3
)2x-xx(1
3y)dydx(x3y)dA(x
:Ans
}x1y2x1,x-1|y){(x,DWhere
3y)dA(xEvaluate1.
:Example
5342
1
1-
44233
1
1-
222222
D
1
1-
x1
2x
22
D
2
2
=+=++=
++++=
+++=
+=+
+≤≤≤≤=
+
∫
∫
∫∫ ∫ ∫
∫∫
+

25. 62xyparabolatheand1-xylinethe
byboundedregiontheisDxydA whereEvaluate2.
2
D
+==
∫∫
36xydxdyxydA
4}y2-1,yx
2
6-y
|y){(x,
}62xy?5,x-3|y){(x,D
:Sol
D
4
2-
1y
2
6-y
2
2 ==
≤≤+≤≤=
+≤≤≤≤=
∫∫ ∫ ∫
+

26. :Sol
2z2yxand0z0,x2y,x
planesby theboundedontetrahedrtheofvolumetheFind3.
=++===
3
1
2y)dydx-x-(22ydA-x-2V
}
2
x-2
y
2
x
1,x0|y){(x,D
D
1
0
2
x-2
2
x
=
==
≤≤≤≤=
∫∫ ∫ ∫所求

27. y}x01,y0|y){(x,
1}yx1,x0|y){(x,D
:Sol
)dydxcos(yEvaluate5.
cos1)-(1
2
1
;)dydxsin(yEvaluate4.
1
0
1
x
2
1
0
1
x
2
≤≤≤≤=
≤≤≤≤=
∫ ∫
∫ ∫

28. Double Integrals in Polar Coordinates
βθαθ ≤≤≤≤= b,ra|){(r,RConsider
Polar rectangle

29. Example ：
3
2
)
3
-
2
()1-(3
2
1
2
3
-
2)1-3(A(R)isRofareaThe
}
23
3,r1|){(r,R3.
}03,r1|){(r,R2.
}201,r0|){(r,R1.
22
22
π
ππ
π
ππ
ππ
π
θ
π
θ
πθθ
πθθ
=
⋅=
⋅⋅=
≤≤≤≤=
≤≤≤≤=
≤≤≤≤=

30. n1,jm;1,i-,r-rrWhere
rr
)r-)(rr(r
2
1
r
2
1
-r
2
1
A
isA-RofareaThe
},rrr|){(r,R4.
1,-jjj1-iii
ji
*
i
j1-ii1-iij
2
1-ij
2
iij
ijij
j1-ji1-iij
 ===∆=∆
∆∆=
∆+=∆∆=∆
∆
≤≤≤≤=
θθθ
θ
θθθ
θθθθ
∫ ∫
∑∑
∑∑
→
∆∆=
∆
= =
= =
β
α
θθθ
θθθ
θθ
b
a
m
1i
n
1j
ji
*
i
*
j
*
i
*
j
*
i
m
1i
n
1j
ij
*
j
*
i
*
j
*
i
)rdrdrsin,f(rcos
r)rsinr,cosf(r
A)sinr,cosf(r
isRonfofsumRiemanuThe

31. ∫∫ ∫ ∫
∫∫ ∫ ∫
=
≤≤≤≤=
=
≤≤
≤≤≤≤=
D
)(h
)(h
21
R
b
a
2
1
)rdrdrsin,f(rcosy)dAf(x,
thenDoncontinuousisfIfregion.
polorabe)}(hr)(h,|){(r,DLet2.
)rdrdrsin,f(rcosy)dAf(x,
thenR,oncontinuousisfIf2-0andrectangle
polarabe}b,ra|){(r,RLet1.
Properties
β
α
θ
θ
β
α
θθθ
θθβθαθ
θθθ
παβ
βθαθ

32. π
θθθ
θθθ
πθθ
π
π
2
15
)d7cos(15sin
)rdrd3rcos)(4(rsin3x)dA(4y
}02,r1|){(r,
4}yx10,y|y){(x,R
:Sol
4}yx10,y|y){(x,Rerewh
3x)dA(4yEvaluate1.
:Example
0
2
R
0
2
1
22
22
22
R
2
=
+=
+=+
≤≤≤≤=
≤+≤≥=
≤+≤≥=
+
∫
∫∫ ∫ ∫
∫∫

33. 2
)rdrdr-(1
)dAy-x-(1V
}201,r0|){(r,D
:Sol
y-x-1zparaboloidtheand
0zplaneby theboundedsolidtheofvolumetheFind2.
2
0
1
0
2
D
22
22
π
θ
πθθ
π
=
=
=
≤≤≤≤=
=
=
∫ ∫
∫∫

34. π
θθ
πθθ
π π
)de
2
1
-
2
1
(limdrdrelim
dAelimdAe
}20n,r0|){(r,DConsider
:Sol
}y-,x-|y){(x,Rewher
dAeEvaluate
:Example
2
0
2
0
n-
n
n
0
r-
n
D
)y(x-
n
R
)y(x-
n
2
R
)y(x-
22
n
22
2
22
2
22
=
==
=
≤≤≤≤=
∞<<∞∞<<∞=
∫ ∫∫
∫∫∫∫
∫∫
∞→∞→
+
∞→
+
+

35. The Cross Product
nofdirectionin thepointsyour thumbthen,btoafromanglethe
throughcurlhandrightyouroffingerstheIf:rulehand-rightby the
givenisdirectionwhoseandbandabothlar toperpendicutorvec
unitaisnand,0,bandabetweenangletheisrewhe
n)sin|b||a(|bavectortheisbandaofproductcrossThe2.
cos|b||a|baisbandaofproductinnerThe1.
vectorsldimensionathreenonzerotwobeba,Let
θ
πθθ
θ
θ
≤≤
=×
=⋅
Definition

36. Example ：
a
θ
bn
.1 .2
a
b
n
ab-ba3.
-jkij,ik(iii)
-ijki,kj(ii)
-kijk,ji(i)
(0,0,1)k(0,1,0),j(1,0,0),i2.
0baifonlyandifparallelarebanda1.
Properties
×=×
=×=×
=×=×
=×=×
===
=×

37. k
bb
aa
j
bb
aa
-i
bb
aa
bbb
aaa
kji
)ba-ba,ba-ba,ba-b(aba
then),b,b,(bb),a,a,(aaIf5.
DbDaD)ba((iii)
Daba)Db(a(ii)
)b(ca)bac(b)a(c(i)
scalarabeacLet4.
21
21
31
31
32
32
321
321
122131132332
321321
3
+==
=×
==
×+×=×+
×+×=+×
×=×=×
.6
a
b
θ
R b |ba|A(R)
isRofareaThe
×=

38. kj-iandjibothtoorthogonalrsunit vectotwoFind4.
baFindk,-jibk,jia3.
k13j-43iba
(2,7,-5)b(1,3,4),a2.
5k3j-6i
021
31-2
kji
ab
5k-3j-6i
31-2
021
kji
ba
(2,-1,3)b(1,2,0),a1.
Example
++
×+=++=
++=×
==
++==×
==×
==

39. Ssurfaceparametricacalledis
RDv)(u,,
v)z(u,z
v)y(u,y
v)x(u,x
such that
Rz)y,(x,pointsallofsetThe2.
t,
y(t)y
x(t)x
:curveparametric1.
:Definition
2
3





⊂∈
=
=
=
∈



≤≤
=
=
βα
Surface Area

40. Rvv,y4,zxhaveweS,z)y,(x,anyFor
surfaceparametricaisS
2sinu}zv,y2cosu,x|z)y,{(x,S1.
:Example
22
∈==+∈
====
x
z
y
)2,0,0(

41. 0)r-(rnplanetheofequationA vector
planetheofvectornormalaisn
z)y,(x,rwhere0)r-(rnbydenotedisplaneThe4.
vectornormalacalledisnvectororthogonalThis
plane.thetoorthogonalisnvectoraandplanethe
in)z,y,(xPrpointabydeterminedisspaceinplaneA3.
:Definition
0
0
00000
=⋅
==⋅
=
x
z
y
r
0r
),,( zyx
n
),,( 00000 zyxpr =

42. 02)-14(z3)-20(y1)-12(x:Sol
R(5,2,0)Q(3,-1,6),P(1,3,2),
pointshethrough tpassesthatplanetheofequationanFind
:Example
(1,2,3)n6},3z2yx|z)y,{(x,S3.
(2,-4,1)n0},z4y-2x|z)y,{(x,S2.
0}4)-z1,y3,-(x(1,2,4)|z)y,{(x,S1.
:Example
=++
==++=
==+=
=+⋅=

43. 
72)
42
2
(cos,
42
2
nn
nn
cos
planesebetween thanglethebeLet
(1,-2,3)n(1,1,1),nareplanestheseofvectorsnormalThe
:Sol
23z2y-xand1zyxplaneebetween thangletheFind
:Example
anglethehavetorsvec
normaltheirifisSandSplanesebetween thangleThe6.
parallelarevectorsnormaltheirifparallelareplaneTwo5.
:Definition
1-
21
21
21
21
≈==
⋅
=
==
=+=++
θθ
θ
θ
θ

44. 9
y
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48. followingin theregiontheisDwhere,xydAEvaluate5.
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x
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xy =
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y =
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49. ∫∫
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54. ksin16cosjsin16sin-icos-16sin
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55. ∫∫
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57. Triple Integrals
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60. dydxz)dzy,f(x,z)dvy,f(x,then
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z)dvy,F(x,z)dvy,f(x,Define
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61. }x-yzx-y-4,yx2,x-2|z)y,{(x,E2.
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z
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22
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128
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x-4
x-4-
4
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62. Riemann-Stieltjes Integral
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)18661826( − )18941856( −
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respect towithf(x)ofintegralStieltjes-RiemannThe
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a
i
n
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i
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63. ∫
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ε
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
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64. ∫∫
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∫
+
++
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4
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x
2
0.5
3
1
x2
0
2
b
a
b
a
i1-ii
p
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1)de(x,1)dcosx(x1.
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b][a,ong(x)to
respectwithfofintegralStieltjes-RiemannThe-f(x)dg(x)1.
:Notation
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choiceanyforand,normawithb][a,ofPpartitionanyfor
2
π
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65. 31)-(220)-(11])[x-]([xtlimxd[x]3.
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4
1
0
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5
136
0
2
)xx
5
3
()dx3x(3xdx3x)1(x2))d(x1(x1.
:Example
(x)dxf(x)g'f(x)dg(x)
thenb],[a,on(x)g'derivativecontinuousahasg(x)where
b][a,ong(x)respect towithintegrableStieltjes-Riemannisf(x)thatSuppose
Theorem9.2
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a
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66. ∫ ∑=
+
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
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a
i
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xc,
cxc,
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g(x)
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λλ
λ
λ
λ
λ
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67. 
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


<≤
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=
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3x1if4
1x0if1
g(x)1.
:Example
1 2 3 4
1
2
3
4
x
y
-152)-(22.54)-(231)-(41dg(x)x(iii)
34)-(421)-(41
))g(x-)(g(xtlimdg(x)x(ii)
-1518-34)-(231)-(41
))g(x-)(g(xtlimdg(x)x(i)
222
3.5
0
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22
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n
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i
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i
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p
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68. [0,1]on1)R(3xe12.
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])[-]([cos2)-(3cos31)-(2cos20)-(1cos1
cosxd[x]cosxd[x]cosxd[x]cosxd[x]cosxd[x]
:Example
f(x)dg(x)f(x)dg(x)f(x)dg(x)f(x)dg(x)1.
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2
3
3
2
2
10
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0
c
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c
b
a
c
a
2
n
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

69. ∫ ∫
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∫ ∫ ∫
=
∈∈
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=+
∈
b
a
b
a
2
0
2
0
222
c
a
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b
a
g(x)df(x)-f(a)g(a)-f(b)g(b)f(x)dg(x)
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π π
π
π

70. incresingisg(x),(x)dg(x)fdg(x)(x)f(x)f(x)f7.
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(x)dg(x)f(x)dg(x)f(x))dg(x)f(x)(f4.
f(x)dg(x)kkf(x)dg(x)3.
f(x)dg(x)k)f(x)d(g(x)2.
g(a)-g(b)dg(x)1.
:Properties
b
a
2
b
a
121
b
a
b
a
b
a
2
b
a
1
b
a
21
b
a
2
b
a
1
b
a
21
b
a
b
a
b
a
b
a
b
a
∫∫
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+=+
+=+
=
=+
=

71. ∫
∫
∫
∫
∫
∫
∫
∫
=
=
+
=+
=+
=+
=
=
⋅=

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4
3
4
1
1
1-
6
0
2
5
0
x
5
0
2
4
6
1
0
2x
b
a
?[x]d[2x]8.
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x
1-e1
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|]dx[|7.
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?xde2.
hf(a)f(x)dg(x)thatShow
bxaifhc
axifc
g(x)b],c[a,f(x)let1.
:Exercise
π
π

72. ∫
∫
∫
∫
∫
∫
∫
∫



=
+
=



>
≤
==
+
→
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0
2
0
2
0
2
1
t
1x
1
1-
0
4
5
4
1
4
5
4
1
exist?xdf(x)Does(iii)
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rationalisxif0
f(x)Let14.
[t])d(telim13.
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2xif2
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g(x)where?cosxdg(x)11.
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x
π