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 :Example

d}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 To

y)}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 into

divided is d][c, and ,x-xx m,1,2,i

]x,[x lsubinterva m into divided is b][a,

n1,2j ; m1,2i ,R

diagonallongest theoflength thedenote |P|Let

ij

m

1i

n

1jij

*ij

*ij

0P||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.

:properties

existslimit 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

*ij

0P||

ww),(o,on continuousnot is f

(0,1)on continuousnot is f

0 x0,

0 x,x

yy)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.

:Example

Ron 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

y3

26

1

3

3

xyydxxA(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-(644

1 (iv)

)dydxy8x-(644

1 (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,dAx1

y1 Find 4.

][0,][0,R where,xcosxydA Find 3.

0-)0

sin2y2

1(-

cos2y)dy-(1dy0

2(-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 If

Theorem) 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,y2

x1 (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 0

Dy)(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 where

d},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.

:Properties

II 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)

22

1-1

2

3

1-

1)x

2

1-xx

2

3x

4

1-x

2

1(

dx4x-x2

33x

2

32x-xx

)dx)(2x-)x((12

3)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

2

6-y

2

2

:Sol

2z2y xand 0z 0, x2y, x

planes by the boundedon tetrahedr theof volume theFind 3.

3

1

2y)dydx-x-(22ydA-x-2V

}2

x-2y

2

x 1,x0|y){(x,D

D

1

0

2

x-2

2

x

所求

y}x0 1,y0|y){(x,

1}y x1,x0|y){(x,D

:Sol

)dydxcos(y Evaluate 5.

cos1)-(12

1 ;)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：

3

2

)3

-2

()1-(32

1

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(r2

1r

2

1-r

2

1A

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

2

15

)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

)de2

1-

2

1(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 vec

unit 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

kbb

aaj

bb

aa-i

bb

aa

bbb

aaa

kji

)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-6i

021

31-2

kji

ab

5k-3j-6i

31-2

021

kji

ba

(2,-1,3)b (1,2,0),a 1.

Example

S surface parametric a called is

RDv)(u,,

v)z(u,z

v)y(u,y

v)x(u,x

such that

Rz)y,(x, points all ofset The 2.

t,y(t)y

x(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 the

in )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 :Sol

R(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)42

2(cos ,

42

2

nn

nncos

planes ebetween th angle thebe Let

(1,-2,3)n (1,1,1),n are planes theseof vectorsnormal The

:Sol

23z2y- 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(uj

u

)v,y(ui

u

)v,x(ur

isector tangent vThe c oSimlarly t (iv)

kv

)v,z(uj

v

)v,y(ui

v

)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

2

1y

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

0

2

1-22

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

kv

zj

v

yi

v

xrk

u

zj

u

yi

u

xr 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

D

vu

x

y

z

0

),,p(

z)y,p(x,

cosz

0 0, sinsiny

cossinx

zyx|op| where 222

4

,cos4

2sin1cossin 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 3.2

1

3cos1z

4

6

4sin

3sin1y

4

6

4cos

3sin1x )

3,

4(1,

coordinate Spherical scoordinater rectangula theFind.2

222222

ksin16cosjsin16sin-icos-16sin

4sin-sin4coscos4cos

0cos4sinsin4sin-

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

D

vu

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 )y

z()

x

z(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

D

vu

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

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x