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Multiple Integral
Double Integrals over Rectangles
Remark:
)( point. sample a called is x 1,2,...n,i ],x,[xx Choose 4.
P of norm }Thex,,x,xmax{ |P| 3.n,1,2,i ,x-xx Define 2.b] , [a ofpartition a called is PThen
bxxxa and }x,,x,{xPLet 1.
ii1-ii
n21
1-iii
n10n10
取樣點
n
1iii
b
a 0P||S of area Thex)xf(limf(x)dx
b] , [aon f of integral definite The 5.
f(x)}y0 b,xa|y){(x,S
[2,3][1,2] 2.4}y2 1,x0|Ry){(x,[2,4][0,1] 1.
Rectangle :Exampled}yc b,xa|Ry) {(x,d] [c,b] [a,R
R rectangle closedA 6.
2
2
Double Integrals and Volumes
V(S) - S of volume thefind Toy)}f(x,zR,0y)(x,|Rz)y,{(x,S d],[c,b][a,RLet 3
lssubinterva into R rectangle theDivide
f(x)dx define toSimilarly b
a
n
1j
m
1iij
*ij
*ij
n
1j
m
1iij
*ij
*ij
ij*ij
*ij
jiijij
j1-ji1-iij
1-jjjj1-j
1-iii
i1-i
A)y,f(x V(S) i.e
A)y,f(x eapproximatcan S of volumeThe
Reach in )y,(xpoint sample a choose
yxA is R of area The
n1,j ; m1,i ]y,[y]x,[xR Define
n1,2,j ,y-yy ],y,[y lsubintervan intodivided is d][c, and ,x-xx m,1,2,i
]x,[x lsubinterva m into divided is b][a,
n1,2j ; m1,2i ,Rdiagonallongest theoflength thedenote |P|Let
ij
m
1i
n
1jij
*ij
*ij0P||
R
R
A)y,f(xlimy)dAf(x,
y)dAf(x, is R rectangle over the f of integral double The
:Definition
R R
R R R
R R
y)dAg(x,y)dAf(x,
thenR,y)(x, y)g(x,y)f(x, If 3.
y)dAg(x,y)dAf(x,y))dAg(x,y)(f(x, 2.
y)dAf(x,cy)dAcf(x, 1.
:propertiesexistslimit thisif
Ron integrable is f then curves,smooth ofnumber finite aon except Ron continuous is f If (ii)
Ron integrable is f then R,on continuous is f If (i)R rectabgle closed on the bounded be fLet
1 Theorem
Rover f of integral double thecalled isy)dA f(x, 2.
exist A)y,f(xlim if R,on integrable is f 1.
:Definition
R
m
1i
n
1jij
*ij
*ij0P||
ww),(o,on continuousnot is f (0,1)on continuousnot is f
0 x0,
0 x,xy
y)f(x, 3.
Ron continuous is f )[0,)[0,R x,y xy)f(x, 2.
Ron continuous is f ][0,2][0,Ry)(x, sinxy,y)f(x, 1.
:ExampleRon continuous is f then R,b)(a, allat continuous is f If 2.
b)(a,at continuous is f then b),f(a,y)f(x,lim If 1.:Definition
2
b)(a,y)(x,
Iterated Integrals
integral iteratedan called is )dxy)dyf(x,(A(x)dx
y)dxf(x,B(y) y)dy,f(x,A(x)Let 1.
d][c,b][a,R y),f(x,function For :Remark
)dyydxx(A(y)dyConsider
y326
13
3xyydxxA(y)Let y, Fixed
b
a
d
c
b
a
d
c
b
a
2
0
3
1
22
0
33
1
2
3
0
2
1
8
0
4
0
2
4
0
8
0
2
3
0
4
1
23
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-(6441 (iv)
)dydxy8x-(6441 (iii)
ydydxx (ii) ydxdyx (i)
evaluate :Example
)dxy)dyf(x,(y)dydxf(x, 3.
)dxy)dyf(x,(y)dydxf(x, 2.
1}y0 2,x-1|y){(x,R where,dAx1y1 Find 4.
][0,][0,R where,xcosxydA Find 3.
0-)0
sin2y21(-
cos2y)dy-(1dy02
(-cosxy)ysinxydxdyBut
?ysinxydydxysinxydA :Ans
][0,[0,2]R where,ysinxydA Find 2.
[1,3][0,2]R wheredA,)3y-(x Find 1.
:Ex
y)dydxf(x,y)dxdyf(x,y)dAf(x,
thed],[c,b][a,Ron continuous is f IfTheorem) s(Fubini' 2 Theorem
R
R
000
2
0
2
0 0R
R
R
2
b
a
d
c
d
c
b
aR
1y)dAsin(xD
thatshow [0,1],[0,1]R If 2.
2y1 4,x3 4,2y1 3,x1 1,1y0 4,x1 2,
y)f(x, 2.
2y0 4,x3 3,2y0 3,x1 2,
y)f(x, 1.
functiongiven theis f wherey)dA,f(x,
Evaluate ,2y0 3,x1|y)(x,RLet 1.Exercises
R
R
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, that Suppose
2}y1 2,x0|y){(x,R [0,2][0,2]R 1},y0 2,x0|y){(x,RLet 3.
1
2
2
2-
1
1-
2
2-
31
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)
dydxx1
y (iv)
xsinydxdy (iii)
dxdyxe (ii)
yxy)f(x, wherey)dydx,f(x, (i)
integrals interated theofeach Evaluate 4.
2}y,13x0 |y){(x,R ,dAx1xy (v)
[0,1][0,1]R ,dAxye (iv)
1}y0 2,x-1|y){(x,R dA,y2x1 (iii)
]3
[0,]6
[0,R y)dA,xcos(x (ii)
3y0 2,x1y)(x,R )dA,3xy-(2y (i)
integral double theCalculate 5.
R
2
R
yx
R
R
R
32
22
1
0
4
0
3
2x
yy
0
x
1
0
1
y
31
0
3
3y
x
1
0
1
0
x2
x-1
2x
0
2
dydxe (vi) dxdy 2ye (v)
dxdyx1 (iv) dxdye (iii)
)dydxy-(3x (ii) dydxcosx (i)
integral interated theEvaluate 7.
4)y1 0,x(-1 |y){(x, rectangle theabove and 15y2x Z
plane under the lying solid theof volume theFind 6.
22
2
1zy xand 0z 0,y 0, xplanes by the Bounded (ii)
2zy
and 2y xcylinders by the Bounded (i)
solidgiven theof volume theFind 9.
1}yx|y){(x,D wheredA,y-x-1 Evaluate 8.
222
222
22
D
22
Din not but Rin is y)(x, if 0Dy)(x, if y)f(x,
y)F(x,
Double Integral over General Regions
function new a Define D.on definedfunction a is f R,D andregion bounded a be DLet
function. continuous twoare h,h whered},yc (y),hx(y)h|y){(x,D
if II typeof be tosaid is Dregion planeA 3.function. continuous twoare g ,g where
(x)},gy(x)g b,xa|y){(x,D if I typeof be tosaid is Dregion planeA 2.
y)dAF(x,y)dAf(x,
is Dover f of integral double The 1.:Definition
21
21
21
21
D R
d}yc (y),hx(y)h |y){(x,D where
y)dxdyf(x,y)dAf(x,
thenDregion II typeaon continuous is f If 2.
y)dydxf(x,y)dAf(x, then
(x)},gy(x)g b,xa|y){(x,D such that Dregion I typeaon continuous is f If 1.
:PropertiesII Type },y1x2y 1,y-1|y){(x,D 2.
I Type 1},ysinx ,x0|y){(x,D 1.:Example
21
D
d
c
(y)h
h
D
b
a
(x)g
g
21
222
1
2
1(y)
2
1(x)
221-1
23
1-1
)x21-xx
23x
41-x
21(
dx4x-x233x
232x-xx
)dx)(2x-)x((123)2x-xx(1
3y)dydx(x3y)dA(x
:Ans}x1y2x 1,x-1|y){(x,D Where
3y)dA(x Evaluate 1.
:Example
5342
1
1-
44233
1
1-
222222
D
1
1-
x1
2x
22
D
2
2
62xy parabola theand 1-xy line the
by boundedregion theis D xydA where Evaluate 2.
2
D
36xydxdyxydA
4}y2- 1,yx2
6-y|y){(x,
}62xy? 5,x-3|y){(x,D
:Sol
D
4
2-
1y
26-y
2
2
:Sol2z2y xand 0z 0, x2y, x
planes by the boundedon tetrahedr theof volume theFind 3.
31
2y)dydx-x-(22ydA-x-2V
}2
x-2y2x 1,x0|y){(x,D
D
1
02x-2
2x
所求
y}x0 1,y0|y){(x, 1}y x1,x0|y){(x,D
:Sol
)dydxcos(y Evaluate 5.
cos1)-(121 ;)dydxsin(y Evaluate 4.
1
0
1
x
2
1
0
1
x
2
Double Integrals in Polar Coordinates b,ra|){(r,RConsider
Polar rectangle
Example:
32
)3
-2
()1-(321
23
-2)1-3(A(R) is R of area The
}23
3,r1|){(r,R 3.
}0 3,r1|){(r,R 2.}20 1,r0|){(r,R 1.
22
22
n1,j m;1,i- ,r-rr Where
rr
)r-)(rr(r21r
21-r
21A
is A - R of area The
} ,rrr|){(r,R 4.
1,-jjj1-iii
ji*i
j1-ii1-iij21-ij
2iij
ijij
j1-ji1-iij
b
a
m
1i
n
1jji
*i
*j
*i
*j
*i
m
1i
n
1jij
*j
*i
*j
*i
)rdrdrsin ,f(rcos
r)rsinr ,cosf(r
A)sinr,cosf(r
is Ron f of sumRiemanu The
D
)(h
)(h
21
R
b
a
2
1
)rdrdrsin ,f(rcosy)dAf(x,
thenDon continuous is f If region. polor a be )}(hr)(h ,|){(r,DLet 2.
)rdrdrsin ,f(rcosy)dAf(x,
thenR,on continuous is f If 2-0 and rectangle polar a be } b,ra|){(r,RLet 1.
Properties
215
)d7cos(15sin
)rdrd3rcos)(4(rsin3x)dA(4y
}0 2,r1|){(r, 4}yx1 0,y|y){(x,R
:Sol 4}yx1 0,y|y){(x,R ere wh
3x)dA(4y Evaluate 1.
:Example
0
2
R0
2
1
22
22
22
R
2
2
)rdrdr-(1
)dAy-x-(1V
}20 1,r0|){(r,D :Sol
y-x-1z paraboloid theand
0z plane by the bounded solid theof volume theFind 2.
2
0
1
0
2
D
22
22
)de21-
21(limdrdrelim
dAelimdAe
}20 n,r0|){(r,DConsider :Sol
}y- ,x-|y){(x,R e wher
dAe Evaluate
:Example
2
0
2
0
n-
n
n
0
r-
n
D
)y(x-
nR
)y(x-
n
2
R
)y(x-
22
n
22
2
22
2
22
The Cross Product
n ofdirection in the points your thumb then ,b toa from angle the throughcurl handright your of fingers theIf :rule hand-right by the
given isdirection whoseand b and aboth lar toperpendicutor vecunit a is n and ,0 ,b and abetween angle theis re whe
n)sin|b||a(|ba vector theis b and a ofproduct cross The 2. cos|b||a| ba is b and a ofproduct inner The 1.
vectorsldimensiona threenonzero twobe ba,Let
Definition
Example:a
bn
.1 .2
a
bn
ab-ba 3. -jki j,ik (iii) -ijk i,kj (ii) -kij k,ji (i)
(0,0,1)k (0,1,0),j (1,0,0),i 2. 0ba ifonly and if parallel are b and a 1.
Properties
kbbaa
jbbaa
-ibbaa
bbbaaakji
)ba-ba,ba-ba,ba-b(aba then),b,b,(bb ),a,a,(aa If 5.
DbDaD)ba( (iii) Daba)Db(a (ii)
)b(ca)bac(b)a(c (i) scalar a bea cLet 4.
21
21
31
31
32
32
321
321
122131132332
321321
3
.6
a
b R b |ba|A(R)
is R of area The
kj-i and jiboth toorthogonal rsunit vecto twoFind 4.ba Find k,-jib k,jia 3.
k13j-43iba (2,7,-5)b (1,3,4),a 2.
5k3j-6i02131-2kji
ab
5k-3j-6i31-2021kji
ba
(2,-1,3)b (1,2,0),a 1. Example
S surface parametric a called is
RDv)(u,,v)z(u,zv)y(u,yv)x(u,x
such that
Rz)y,(x, points all ofset The 2.
t,y(t)yx(t)x
:curve parametric 1.
:Definition
2
3
Surface Area
R vv,y 4,z xhave weS,z)y,(x,any For
surface parametric a is S 2sinu}z v,y 2cosu,x|z)y,{(x,S 1.
:Example
22
x
z
y
)2,0,0(
0)r-(rn plane theofequation A vector plane theof vector normal a isn
z)y,(x,r where0)r-(rnby denoted is plane The 4. vectornormal a called is n vector orthogonal This
plane. the toorthogonal is n vector a and plane thein )z,y,(xPrpoint aby determined is spacein planeA 3.
:Definition
0
0
00000
x
z
y
r0r
),,( zyx
n
),,( 00000 zyxpr
02)-14(z3)-20(y1)-12(x :SolR(5,2,0) Q(3,-1,6), P(1,3,2),
points he through tpasses that plane theofequation an Find :Example
(1,2,3)n 6},3z2yx|z)y,{(x,S 3. (2,-4,1)n 0},z4y-2x|z)y,{(x,S 2.
0}4)-z1,y3,-(x(1,2,4)|z)y,{(x,S 1. :Example
72)422(cos ,
422
nnnncos
planes ebetween th angle thebe Let (1,-2,3)n (1,1,1),n are planes theseof vectorsnormal The
:Sol23z2y- xand 1zy xplane ebetween th angle theFind
:Example angle thehave tors vec
normal their if is S and S planes ebetween th angle The 6. parallel are vectorsnormal their if parallel are plane Two 5.
:Definition
1-
21
21
21
21
9y
y)f(3, 2x,x
f(x,1) y,xy)f(x,z
:Example
2
1L2L
)1,(:1 xfzC
),3(:2 yfzC
x
y
z
ii C of line tangent theis 1,2,i ,L
0rr ifsmooth called is S surface The (v)
k u
)v,z(uju
)v,y(uiu
)v,x(ur
isector tangent vThe c oSimlarly t (iv)
kv
)v,z(ujv
)v,y(uiv
)v,x(ur
obtained is )v,z(uz ),v,y(uy )v,x(u x where)z,y,(x-Pat c ector to tangent vThe (iii)
Son lying c curve grid a defines andu parameter single theoffunction vector a is )vr(u, (ii)
Son lying c curve grid a defines and vparameter single theoffunction vector a is v),r(u i)(
)v,(u Fixedv)k z(u,v)jy(u,v)ix(u,v)r(u, v)}x(u, xv),y(u,y v),z(u,z|z)y,{(x,S by defined be and surface parametric a be SLet
:Definition
vu
000000u
2
000000v
000000
00000001
2
0
1
0
00
Exercises 2
page ofout of page theinto directed isu whether vdetermine and |vu| Find 2.
(0,2,4)b (1,2,0),a (iii) (5,-2,-1)b (1,2,-3),a (ii) (3,0,1)b (-2,3,4),a (i)
ab ,baproduct cross theFind 1.
5|| u6|| u
6|| u
)2,0,1(u
5|| v
8|| v
)2,1,1(v8|| v
60 60
150
y)f(x,lim Find (ii) (0,0)at continuousnot is y)f(x, that show (i)
(0,0)y)(x, if, 1
(0,0)y)(x, if ,yx
xyy)f(x,
function heConsider t 4.not?or why why (0,0)?at continuous f Is
(0,0)y)(x, if, 0
(0,0)y)(x, if ,yx
yxy)f(x,
function heConsider t 3.
o)(o,y)(x,
22
24
2
following in theregion theis D where,xydA Evaluate 5.D
D
x
y2xy
21y
2yx
D
222 4}yx1|y){(x,D where)dA,sin(xy Evaluate 6.
D
2
2222
D
2
dAxy Evaluate 8.
4y xcircle theand 42y x
ellipse ebetween thregion theis D wheredA,x Evaluate 7.
D2
4 (2,4)
0
Sin
0 0
Siny
0
22
0
Cos-1
0
2
0
Cos
0
2
dxdy x (iv) drdr (iii)
drdrSin (ii) drdSinr (i)
integrals iterated theEvaluate 9.
34
43
5Sec-
0
23
222
D
22
D
22
2
1
x-2x
021-22
1
0
y-1
0
22
22
D
yx
drdSinr
evaluate then and scoordinater rectangula Switch to 12.4yx|y)(x,D
wheredA,|yxSin| ,yxSin (iii)
dydx)y(x (ii)
)dxdyySin(x (i)
Evaluate 11.4y x
by enclosedregion theis D where,dAe Evaluate 10.
2
2
22
3
0
x-9
0
x
0
1
0
2x
x
yx
0
0
2
0
z-4
0
1
0
z
0
y
0
3
22
22
2
2
2xydzdydx (iv) yzdydzdx (iii)
y zsinydxdzd (ii) xyzdxdydz (i)
integral iterated theEvaluate 16.9x4 ,xy (ii)
2x0 ,xy (i)
axis- xabout the curvegiven therotatingby obtained surface theof area theFind 15.1vu v,-uz v,uy uv, xsurface theof area theFind 14.
9z plane thebelow yxz surface theof area theFind 13.
),,( coordinate Spherical 1.:Example
kvzj
vyi
vxrk
uzj
uyi
uxr where
dA|rr|A(S) is S of area surface then theD,domain
parameter t the throughouranges v)(u, as oncejust covered is S and Dv)(u, v)k,z(u,v)jy(u,v)ix(u,v)r(u,
equation by thegiven is and surface parametricsmooth a be SLet 1.Definition
vu
Dvu
x
y
z
0
),,p(z)y,p(x,
cosz0 0, sinsiny
cossinxzyx|op| where 222
4 ,cos
42sin1cossin x
4 ,2cos2cosz
2)2(11zyx (i)
:Sol,-2)3(0,2 (iii) ,0,1)3( (ii) )2(1,1, (i)
scoordinate spherical r torectangula from Change 3.21
3cos1z
46
4sin
3sin1y
46
4cos
3sin1x )
3,
4(1,
coordinate Spherical scoordinater rectangula theFind.2
222222
ksin16cosjsin16sin-icos-16sin
4sin-sin4coscos4cos0cos4sinsin4sin-kji
rr
k)4(-sinjsin4cosicos4cos r k0jcos4sinisin-4sinr caculate
}0 ,20|),{(D),( and k4cosjsin4sinicos4sin ),r(
isequation parametric The :Sol S of area surface theFind (ii)
4 radius of spherical a of surface The -S
20 ,0 4cosz ,sin4siny ,cos4sinx|z)y,(x,
S (i)
S surface parametric .4
22
Dvu
D0
2
0
22222
dA|rr|A(S) 1.
Remark
164dd16sindA|rr|A(S)
16sin
)sin(16cos)sin(-16sin)cos(-16sin|rr|
Thus
u
v
v
uijR
),( vur
x
y
z
urvr
ijS
y)k)f(x,yjxiy)r(x, (
dA )yz()
xz(1A(S)
is S of area surface The D}y)(x, y),f(x,z|z)y,{(x,S 2.
dArr
vu)r(r)A(SA(S)
vu|rr||rvru|)A(S
S of area The
D
22
Dvu
vuij
vuvuij
ij
Triple Integrals
z width equal of ]z,[[z lssubinterva
n into divided a is f][e, y, width equal of ]y,[[y lssubinterva m into divided a d]is[c, x, width equal of ]x,[x lssubinterva l
into divided is b][a, box.r rectangula a be f][e,d][c,b][a,BLet :Definition
[0,2][1,3][0,1]B 1.:Example
f][e,d][c,b][a, f}ze d,yc b,xa|z)y,{(x,B
:boxr Rectangula
k1-k
j1-j
i1-i
existslimit thisif
v)z,y,f(xlimz)dvy,f(x,
is Bbox over the f of intergral tripleThe 4.
v)z,y,f(x sumRiemann tripleThe 3.
zyxv B of volumeThe 2.
]z,[z]y,[y]x,[xB 1.
*ijk
*ijk
l
1i
m
1j
n
1k
*ijk
B0nm,l,
*ijk
*ijk
l
1i
m
1j
n
1k
*ijk
ijk
k1-kj1-ji1-iijk
?yzdvx define toHow
y}-x-1z x2,y0 1,x0|z)y,{(x,E 3.
[0,2][1,3][-1,1]B where,yz)dv(x Evaluate 2.
[1,2][1,2][0,1]B where,dvxyz Evaluate 1.
:Example
z)dxdydzy,f(x,z)dvy,f(x,then
f][e,d][c,b][a,Bon continuous is f If
Theorem) sbini'Theorem(Fu
E
2
B
B
2
B
f
e
d
c
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