第十章重積分試題庫(kù)(無(wú)水印)

、 知識(shí)點(diǎn)(第十章重積分)0101二重積分的概念010101二重積分的定義010102010103二重積分的物理意義0102重積分的性質(zhì)010201二重積分的性質(zhì)010202二重積分的對(duì)稱(chēng)性0103二重積分的直角坐標(biāo)計(jì)算法010301用直角坐標(biāo)計(jì)算二重積分010302交換積分次序0104二重積分的極坐標(biāo)計(jì)算法010401二重積分化為極坐標(biāo)系下二次積分三重積分0201三重積分的概念020101三重積分的概念020102三重積分的性質(zhì)020103三重積分的對(duì)稱(chēng)性0202三重積分的計(jì)算用直角坐標(biāo)計(jì)算三重積分用柱面坐標(biāo)計(jì)算三重積分用球面坐標(biāo)計(jì)算三重積分0301幾何應(yīng)用030101計(jì)算曲面的面積030102計(jì)算立體體積0302物理應(yīng)用030201計(jì)算物體質(zhì)心030202計(jì)算轉(zhuǎn)動(dòng)慣量030203計(jì)算引力[100101][][0.2][][ ][][

f(x,y)DDii,n)

),m

f(,)存i i i

0

i1

i i i(是 )

f(x,y)Df(x,y.D[] i,n)。i[100102][][0.2][][

f(x,y)DDnii,n),

選),m

f(,)i i

0

i1

i i i(i。

i,n) []fxyD上[100103][][0.2][何知識(shí)2][][]D面S= .D[]2S.[100104][][較0.3][何含][][何含,坐系下][]D面S,坐系下Drdrd= .D[]S.[100105][][較0.3][何][角坐系下計(jì)算][計(jì)算][]D(0,0)))三角形區(qū)域計(jì)算二重積分xy

=___________.D[答案評(píng)分標(biāo)準(zhǔn)]16.[100106][填空][易0.2][二重積分物理意義][二重積分計(jì)算][二重積分計(jì)算,二重積分物理意義][]D其上(xy處面密度(xy如果(xy)在D上連續(xù)則薄片質(zhì)量m= _.[答案評(píng)分標(biāo)準(zhǔn)](xy)d ((x).D D[100107][填空][較易0.3][二重積分幾何意義][二重積分計(jì)算][二重積分幾何意義,二重積分計(jì)算]根據(jù)二重積分

1x2

y2d

=___________ 其中D：x2y2

D1.[答案評(píng)分標(biāo)準(zhǔn)]2.3[100108][填空][較易0.3][二重積分幾何意義二重積分概念][]

f(t連續(xù)函數(shù)z0x2y2

1zf(xy)]2所圍立體體積可用二重積分表示_.[答案評(píng)分標(biāo)準(zhǔn)]x2y2

[f(xy)]2dxdy[100109][填空][較易0.4][二重積分幾何意義][][二重積分,極坐標(biāo)系下二重積分計(jì)算][]

D:0r1,02

，1r2rdrd= _.D[答案評(píng)分標(biāo)準(zhǔn)]16][][]

fx,yDf(x,y)0,

f(x,y)d

D[]z

f(x,y)D。][0.3][計(jì)算][][]D:0y

a2x2,0x

知 a

x2

y2dxdyD= .[]1a36][較0.3][][計(jì)算][][]Dx2y2

2,由

2x2

y2dxdy= .D[]4 234 23][較0.4][][計(jì)算][][]Dx2y2

2x由

2xx2

y2dxdy= .D[]2.3][較0.3][對(duì)稱(chēng)性][計(jì)算對(duì)稱(chēng)性][]Dx2y2

4y0,x3y2

= .D[0][較0.3][對(duì)稱(chēng)性][計(jì)算對(duì)稱(chēng)性][]

fx,yy軸對(duì)稱(chēng)D

f(x,y)f(x,y)，則f(x,

=__________.D[0][0.3][][][]D:x2y2a2,y0m為奇數(shù)時(shí)， xmyn= _.D[0][0.3][][][]D0xaa

ya,n

xmyn= _.D[]0][0.4][交換][][交換次序]f(x,y)1y10 y

f(x交換次序后為_(kāi)______________.[]1x0 0

f(x,y)dy.][0.4][交換][][交換次序]f(x,y)1yy0 y

f(x,

交換次序后為_(kāi)_____________.[]1x

f(x,y)dy0 x2[100120][][0.4][交換][][交換次序][]f(x,y)為連續(xù)函數(shù)，次axx0 0

f(x,y)dy

交換次序后為_(kāi)________________.[]ay

f(x,y.0 y[100121][][0.4][][][][]f(x,y)2xx20 0

f(x,y

.[]4y20

f(x,y[100122][][0.4][][][][]f(x,y)1xx

f(x,y

.[]

0 x21y0 y

f(x,yx2[100123][][0.5][下x2[]

f(x,y

在下先對(duì)r為 .[]

1 00 0

f(rcos,r)rdr[100124][][0.5][下][][下]aa

dx

a2x2a2x

f(x,y

在下先對(duì)r .[]20 0

f(rcos,r)rdr.[編號(hào)][][0.5][下][][下][]1x0 0

x

f(x,y

在下先對(duì)r為 .[]2d20 0

f(rcos,r)rdr[][][0.7][][][][]Dx2y

1x2y

2x

f(xyDr.3[]33d3

f(r,r)rdr +

d

f(rcos,r)rdr + 02

032d2

f(r,r)rdr 03[][][0.3][][][][]fx[04上連續(xù)Dx2y2

4

f(x2

y2

Dr.[]2d0 0

f(r2[][][0.3][幾何意義][][計(jì)算,幾何意義][]

4x2

y2dxdy

= D:x2y

Dy0.[]4.3[][][0.3][幾何意義][][計(jì)算,幾何意義][]根據(jù)幾何意義 a

x2

y2dxdy =

Dx2y

a2

y

a0.D[]1a3.3[][][0.3][][][,][]D0

x

0

xxD

y2

= .[]13[][][0.4][][][,][]。則質(zhì)量公式為 .[]M=[][][中等0.6][][直線][,][]D0xx yy。則關(guān)于直線 0

0z

為 .[]

0 0|AxByCzD|Ixx)2(yy)2] 0 0

B2

C2[][][0.35][對(duì)稱(chēng)性][][]

(ey2|y|1

sin

y3z2x3)dv

則I= 。[]I24[][][0.35][對(duì)稱(chēng)性][][]

x2y2z1

[x3ezx2)

= 。[答案及評(píng)分標(biāo)準(zhǔn)]I4[編號(hào)填空易0.2][重積分的性質(zhì)三重積分][]

f(x,y,z)在有界閉區(qū)域上可積， ，則2I

f(x,y,z)dv

f(x,yz)dv。 1[答案及評(píng)分標(biāo)準(zhǔn)]2 1

f(x,y,z)dv[編號(hào)填空易0.2][重積分的性質(zhì)三重積分][]

(x23y2

(3xy1x2)dv2 [答案及評(píng)分標(biāo)準(zhǔn)]I(3xy3x23y2)dv2[編號(hào)填空][0.4][重積分的物理應(yīng)用三重積分]設(shè)(x,y,z)設(shè)M(x,y,z)為其重心，關(guān)于xoy平面的靜矩定義為：Mxy .[答案及評(píng)分標(biāo)準(zhǔn)]

,

M 的三重積分計(jì)算式為xyM xy

[][0.4][,][]x2

y2z2

R2

z

f(t((C)

f(x0f(x0

f(xf(x

0

f(x)dv [A[][][0.35][][][][]:x21

y2z2

R2

:x22

y2z2

R2

x

y

z0.u

f(t)是((0(A)

xf(x)dv4xf(x)dv

(B)

f(x

f(xz)dv4 4(C)

f(x

y)dv

2

f(x

y)dv

(D)

1

f(xyz)dv

4

f(xyz)dv 1 2 1 2[](D)[][][0.2][][D

f

mn0i

f,i i

A ； B C ； D 。[]D[][][0.2][][]

x1i n

y12jn

,

j,n

域D:1

x1

y3割成一系列方形(x2y2)dmnni1

nj

D12;nn12;nn )2 )2]n n

n

)2

j 12i)2] ;in

i

j

n n nnmnm

ni1ni1

i 1 1 )2 ; n n n )2n n n[]A[][][0.2][][][][]f(xDAfx,yBDx,yCfDD fD。[]C[][][0.2][][][][]x

i,yn

j,(i,n

jnD0

x1,0

y1割成一系列小正方形則

xydxdy

n i

Di1n

i1

n n n2mn

nnj

ii1n n n2

n i1i1n

i1

n n n2D

nn(i

i)11n

i1

n n n n[]B[][][0.2][][][][]f(x,y)Df(x,y

DA ； B C D 。[答案及評(píng)標(biāo)準(zhǔn)]B[][][0.2][][][][]f(x,y)Df(x,y)d

DA ； B C D 。[答案及評(píng)標(biāo)準(zhǔn)]C[][][0.3][直角坐標(biāo)系下計(jì)算][]xydx

(D0

x2,0

1)值為D1 1 1 1A B C D6 12 2 4[答案及評(píng)標(biāo)準(zhǔn)]B計(jì)算][對(duì)稱(chēng)性][計(jì)算][]若區(qū)D為0

x2,|

2,則xy2dx=A 0 B 323

DC 64 D 3[答案及評(píng)標(biāo)準(zhǔn)]A計(jì)算][性質(zhì)][計(jì)算][]設(shè)

fxyx2y

1使x2y2

f(x,y)dx

41x0

x

f(x,y)dy成立f(x,

f(x,

f(x,y)f(x,y)f(x,

f(x,

f(x,

f(x,y)C f(x,f(x,f(x,f(x,D f(x,f(x,

f(x,

f(x,y)[]B[][][0.3][][][][]Dxoyxy1f域1D:xy1f(2,2)y

f(x2,y2DDD1A 2 B 4 C 8 D 12[]B[][][0.3][][][]f(,y)exnx1 0

f(x,y)dy次序結(jié)果為eyn1 0

f(x,y)dxeey

y1f(,)x0nxy

f(x,y)dx1y

f(x,y)dx0 1[]D

0 ey[][][0.3][][][]f(,y)1yy322xA 1xx32y2yB0 0yx1322y0 0C 1x232y2yD0 01x232y2y0 00 0[]C]f(x,y)axx0 0

f(x,y)dy

(a0)ay

f(x,y)dx

ay

f(x,y)dx0 0 0 aayaf(x,y)x D ay

f(x,y)dx0 y 0 0]C]f(x,y)1xx0 0

f(x,y

xy1f(x,y)x0 0

1y0 0

f(x,y)dx1y1f(x,y)x0 0

1y0 0

f(x,y)dx]D]f(x,y)0x

x

f(x,y)dy=1yy1f(x,y

2y

1y2f(xy2

x10 1 1 11yy1f(x,y)xy2y21yy1f(x,y)x

2y

f(x,y)dx0 1 y22y2

f(x,y)dx0 1]C][]f(x,y

D:y

x

yx

分f(x,y

可化累D0xx

f(x,y)dyx0xx

f(x,yx1dyy

f(x,y)dx0 y1dyy

f(x,y)dx0 y[]C[][][0.5][下二重積的計(jì)算二重積][內(nèi)容]

f(x,y)1y0

3yy22

f(xy)dx可交換積次序?yàn)?x

2xf(x,y)dy

3

3x2

f(x,y)dy0 0 1 021x2

2xf(x,y)dy

2x1f(x,y)y

3

3x

f(x,y)dy0 1x

3x

12f(x,y)dy

0 2 00 2xD 3

f(rr)rdr2 2cos0[]B[][][0.5][下二重積的計(jì)算二重積][內(nèi)容]

fxy為連續(xù)函數(shù)，則積分1xx2

f(x,y)y2x2

f(x,y)dy0 0 1 0可交換積次序?yàn)?y

f(x,y)x2y2

f(x,y)dx0 0 1 01yx

f(x,y)x2y2

f(x,y)dx0 0 1 01y2

f(x,y)dx0 y1y2

f(x,y)dx0 x2[]C[][][較易0.4][極下二重積的計(jì)算二重積的計(jì)算][]D(x1)2y

1,

f(x,y)dxdy

0 0

Df(rr)rdr22

2co02co0

f(rr)rdrf(rrsin)rdr2D 2d2co20 0

f(rcos,r)rdr[]C[][][0.4][][][][]Dx2y

2x(xy

x2y2dxdyD22con) 2r2 02n)2cor30 022in)d2cor3dr0 022n)2cor3 02[]D[][][0.4][][][][]Dx2y

1,fD上連續(xù)函數(shù)f(

x2y2)dxdy=D1f(r)01f(r)0C 1f(r2)0D rf(r)0[]A[][][0.4][][][][]I1

xD1

I (x2D

I sin7(x,其3DD

x0,y0,

xy ,

xy1I,I,III I1 2 3

2I I I3 2 1

1 2 3II I1 3 2

I II3 1 2[]C[][][0.4][][][][]2

dxdy1x

IyA I31

B 2I3C 0I D2

1I0[]A[][][0.4][][][][]I1

xy)dD1

I (xy)2d2D

I (xy)dD3Dx0,

y0,

xy xy1I,I,IA I I I3 2 1

2B II I1 2 3

1 2 3II I1 3 2

I II3 1 2[]B[][][0.4][][][]

( )[]DDoyD

f(x,y1 2 1 2D D上連續(xù)函數(shù)D1 2D

f(x2,A

f(x2,y)dxdy

B

f(x2,D D1 2C D1

f(x2,y)dxdy

D 12D2

f(x2,y)dxdy[]A[][][0.4][][]D1,1,)=A e

De1C 0 D []C[][][0.4][][]Dx2y2

a2

(a)a

a2x2y2dxdyD3234323412A 1 B C D[]B[][][0.4][][]0

x1;0

y1;0

z1

f(xyz

f(xyz。m

f i i i 13

n

i i i 1n

i1

( , , )( )nn n n

n

i1

f( , , )nn n nm

n

f i j k 13

nn

i j k 1n

i1

jk

( , , )( )nn n n

n

i1

j

k

f( , , )nn n n[]C[][][0.4][][]0

x1;0

y1;0z1

f(xyz有界函數(shù)。若m

n

f i j k 13n

i1

jk

( , , )( nn n n

I則A f(x,y,z)積 B f(x,y,z)一定

f(xyzI0

f(xyz必試題答案及評(píng)分標(biāo)準(zhǔn)]B]試題內(nèi)容F(xyz有界閉域(xy

f(x,y,

(x,y,z)，則：1 2F(x,y,f(x,y,f(x,y,1 2 A 式成立 B 式成立C f(x,y,時(shí)成立 D1

f(xy也未必成立1試題答案及評(píng)分標(biāo)準(zhǔn)]C]試題內(nèi)容設(shè),是空間有界閉區(qū)域，

f(xyz, , 2

f(x,y,

f(x,y,

f(x,y,z)dv的充要條件是 3 1 2A f(x,y,z)是奇函數(shù) B4

f(x,y,z)0

(x,y,4C 4

D

f(x,y,z)dv0試題答案及評(píng)分標(biāo)準(zhǔn)]D

4答( )]試題內(nèi)容設(shè)f(xyz是一全空間的連續(xù)函數(shù)，由中值定理

f(x,y,

f(,,V.(,,而V為的體，則：A f(xyzxyz為奇函數(shù)時(shí)B f(,,)0

f(,,)0若x2y2

1

f(,,)

f(0,0,0)f(,,xyz的奇偶性無(wú)必然聯(lián)系試題答案及評(píng)分標(biāo)準(zhǔn)]D[][][0.3][三重積分的性質(zhì)三重積分的性質(zhì)]

( )[內(nèi)容]設(shè)uf(t在(是上半單位x2y2

1,z0,

I

f(xy,則A I0

B I0I0

I的符不定[案及評(píng)分標(biāo)準(zhǔn)]B[][][0.3][三重積分的性質(zhì)三重積分的性質(zhì)][內(nèi)容]設(shè)u

ft)是(,)|x1,|y1,|z1

I

f

a,b,c為常數(shù)，則I0

I0I0 D I的符由ac確定[案及評(píng)分標(biāo)準(zhǔn)]C[][][0.3][三重積分的性質(zhì)三重積分的性質(zhì)]

( )[內(nèi)容]設(shè)uf(t是(上嚴(yán)格單調(diào)減少的奇函數(shù)，I x2y2z2A I0

kf(xyz

k0B

I0C I0 D 當(dāng)k0I0;當(dāng)k0I0[案及評(píng)分標(biāo)準(zhǔn)]A[][][0.3][三重積分的性質(zhì)三重積分的性質(zhì)]

( )內(nèi)容為單位球體

x2y2

1

位于1

z

部分的半球體，I(xyf(x2y2z2,則I0

I0I0

D I(xyf(x2y2z21[案及評(píng)分標(biāo)準(zhǔn)]C[][][0.3][][][][]x2y2I

1,

f(x,y,)

x2(x,y2,,A 4 x2y2z2y0,z0

x2(x,y2,z3

B 4 x2y2z2x0,y0

x2yzf(x,y2,z3)dvC 2 x2y2z2z0

x2(x,y2,z3D 0[]D[][][0.4][][][][ ] Ie1

x2y2z2dv ,

I x2y2z2,2I 3

x2y2z2,z

x2y2

x2y2

1I,1

I, I2 3A. II1

I; B3

II1

I; C2

I II; D.I I2 1 3 3

I.1[]B[][][0.4][][][][]1

:x2y2

R2

z0;2

:x2y2

R2;x0,y0,z0.則A dvx99dv

. B y99dvdv Cx9v4y9C

1()9v 4() . . 99 1 2[]A

1 2( )[][][0.5][][][][]x0

y0

z02xyz1

則

f(x,y,z)dvA 1y1x2x0 0 0

f(x,y,2B 1yyx2x20 0 0

f(x,y,z)dz2C 1y1x20 0 0

f(x,y,z)dz2D 1dz1dx220 0 0

f(x,y,z)dy]B]3x2y2

z

z1x2

f(xyz1414z2

f(xyzy23zy23zy23zy23Af xyz

(, , )

21

(, , )1y2B2dx dy f xyz0 z 1y2B2dx dy f xyz14z214z221x y3x2y2f(,14z214z22

y y2 f(,y,)1 2

1z2

2

3x2y22121y2D( )]zx2y2, yx, y0,z1一卦限部f(x,y,z)A1A1yy2x1 f(x,y,)Bdx 22 yy10yx2y20yx2y2

f(xy2

f(x,y,z)dzC dy 22 yx1 C dy 22 yx1 f(,y,D22dy1y2x10yx2y20y0]C]

( )x2y2

2z,

zx2y2確定立體體A 1r

1r2dz

B rr

1r2dz0 0 r2 0 0 11r2C 1rr2 z D 1rr21r20 0 []C

0 0 r2[][][0.5][三重積化為三次積三重積的計(jì)算][內(nèi)容]設(shè)x2y2(z于

f(t

f(x2y2z2)dvA d1f(r2)r2n0 0 0C d1f(2rs)r2n0 0 0

Bd1f(r2rs)r2n0 0 02D2d1f(2rs)r2sindr20 0 0[]Bx2y2[][][0.5][三重積化為三次積x2y2[內(nèi)容]設(shè)是由1x2y2

4; z

2 2

f(z)dv于2 A 4d2 0 0 1

f(r)r2sindr

B d0 0 1

f(rcos)r2sindr2C 2d0 0 12

f(cos)r2sindr

D 2d20 0 2

4r2

f(rcos)r2sindr[]A[][][0.4][][][][]

f(xyxy2D0

1, 0

1。[答案及評(píng)準(zhǔn)]D

f(x,y)dx

1xx1y2y1 50 0 6而D當(dāng)面1, 71f(x,y)D.6 10[][][0.2][次][][次]3y2(x2)x1 1[答案及評(píng)準(zhǔn)]1原式(31)(3

x3x)2 712(78 3 3[][][0.2][次][][次]4x2

ydy.[答案及評(píng)準(zhǔn)]43xdx

2 x x522=9. 10[][][0.2][次][][次]2ynyexx.1 0[答案及評(píng)準(zhǔn)]21

1)dy1 102[][][0.2][次][][次]2 2 []x1 0

xydy.[答案及評(píng)準(zhǔn)]原式

221dx 531x22 103[][][0.2][次][][次]axxy.0 0[]a 50a2 a 10a23[][][0.2][下二次積的二次積的]2y2x.0 0[]2y2x 50 0=4. 10[][][0.2][下二次積的二次積的]9x41 0

xydy.[]9

xdx4

51 0832= . 109[][][0.2][下二次積的二次積的][內(nèi)容]

x

y4dy20 cosx[]220

151dx 5=8. 1010 75[][][0.3][下二次積的][][二次積的]xsxy2ny0 0[]1n1s)3x 5304= . 103[][][0.3下二重積的二重積的][內(nèi)容]

ysyx2n

ydx2 02[]2922

sin

ycos3

5=12. 105[][][0.3][下二重積分的][][二重積分的][內(nèi)容]D

11y2

d,Dx2,|y1.[答案及評(píng)分準(zhǔn)]2x1 1 y 52 11y2=42arctan1.. 10[][][0.2][下二重積分的二重積分的][內(nèi)容]

D:0

x1,0

y2.D[答案及評(píng)分準(zhǔn)]1x2y 40 01x2y. 70 0=1 10[][][0.2][下二重積分的二重積分的][內(nèi)容]

,

D:0

xa,0

yb.D[答案及評(píng)分準(zhǔn)]=a

dxb

50 024(ab)3 29[][][0.2][下二重積分的二重積分的][內(nèi)容]D

y d,D01x

x1,0

y2.[答案及評(píng)分準(zhǔn)]1 1 x2y 501x 02ln2 [][][0.2][下二重積分的二重積分的][內(nèi)容]ex,

D:0

x1,0

y1.D[答案及評(píng)分準(zhǔn)]1exx1eyy 50 0(e[][][0.2][下二重積分的二重積分的][內(nèi)容]D

x21y2

,D0

x1,0

y1.[答案及評(píng)分準(zhǔn)]1x2x1 1 y 50 01y218 1012[][][難程度][1][關(guān)鍵詞][][][0.2][下二重積分的][][二重積分的][答案及評(píng)分準(zhǔn)]1x2x2y 51 04 3[][][0.2][下二重積分的二重積分的][內(nèi)容]D

x 1y2

, D0

x1

y1.[答案及評(píng)分準(zhǔn)]2x1 1 y 40 11y222arctan1 7 10[][][0.2][下二重積分的二重積分的][內(nèi)容]D

,

D:0

x,0

y .2[答案及評(píng)分準(zhǔn)]2sinxsy 520 02 [][][0.2][下二重積分的二重積分的][內(nèi)容]分,中D:1

x3,0

y2.y1D[答案及評(píng)分準(zhǔn)]式3x2x2 1 y 51 01y28ln3 103[][][0.2][下二重積分的二重積分的][內(nèi)容]

,D0x1,0

y4.D[答案及評(píng)分準(zhǔn)]式13x4y 50 03 10[][][0.2][下二重積分的二重積分的][內(nèi)容]xsinD

D:1

x2,0y2.[答案及評(píng)分準(zhǔn)]2原式2dx21 0

53 102[][][0.3][下二重積分的二重積分的][內(nèi)容](x

d

,

D,|y1.D[答案及評(píng)分準(zhǔn)]原式0

x1

y1

yx 510[][][0.3][下二重積分的二重積分的][內(nèi)容]x(x,

D3,|y1.D[答案及評(píng)分準(zhǔn)]原式3

x2x1

y3

x1

510[][][0.2][下二重積分的二重積分的][內(nèi)容](x3

,

D1,0y1.D[答案及評(píng)分準(zhǔn)]

3x1y

x1y2y1 023

1 0[][][0.2][][][][],DO(0,0),D。[答案及評(píng)準(zhǔn)]1x1y 40 x11)x 701 6[][][0.2][][][][],

D:0x1

y0.D[答案及評(píng)準(zhǔn)]1x

40 111x)x 701 e[][][0.2][][][][](xy2

D:0

yx,0x.D[答案及評(píng)準(zhǔn)]xnx(xy2)y 40 0(xinx1in3)dx 70 34 109[][][0.3][][][][],D由曲線y,線y0, x2,所圍成D。[答案及評(píng)準(zhǔn)]2x0 0

412dx 72 016 103[][][0.3][][][][]

,Dyx,

y2x

x4。D[答案及評(píng)準(zhǔn)]原式4x2x

xydy 4043x2

xxdx 702384 107[][][0.3][][][][],

D:xy

x2.D[答案及評(píng)準(zhǔn)]原式2x

3x41 x2x3dx 7133 4[][][0.3][][][][]xD是線x0,y和y

x。原式x(x

Dy)dy 30 x(x)n2)x 702 [][][0.3][][][][](x2y2,D是線yx,

yx1,

y1,Dy3。[答案及評(píng)準(zhǔn)]原式3y

(x2y2y)dx 41 y13[11 3

y3(y1)3)

y2

y]dy3[2y22y1]dy 71 310 10[][][0.3][][][][]xcos(2,D

D:0x4

,1

y1.[]4x4

xcos240 14sin27401 102[][][0.2][][][][]ex,

D:1x1

y1.D[]11

exx111

eydy 5(e )2 e[][][0.2][][][][](2,

D:|x,0

y1.D[]2x1y 5 00[][][0.2][][][][](x,Dyxx0,y1D[]1yx(x0 0

y)dx 411(x02

y)2|ydy01(2y21

y2)dy 70 21y312 01 102[][][0.2][][][][](x6,Dyx,

y5xx1D。[答案評(píng)準(zhǔn)]1x5x(x6)y 40 x16x2x 70251 103[][][0.3][][][][],D

y1

yxx2D。[答案評(píng)準(zhǔn)]2xxy

x4121

1xx(x2

1)dx 712 x2151ln2 8 2[][][0.2][][][][]D

ydxdy,Dyx

xx2x4。[答案評(píng)準(zhǔn)]41dx2xdy 42x x437229 [][][0.3][][][]yy,Dxy1.D[答案評(píng)準(zhǔn)]41xxy 40 0211x2)x 702 3[][][0.3][]yd,Dxy1.D[]41xxy 40 0211)2x 701 6[][][0.3][][][][],D:1x1x

yx.D[]2xxy2y 41 1x12(x41)dx 731 x219 1010[][][0.3][][][][]D

1(x

,D:3x

y2.[]4x3 1

(x

dy 3y)243

1 x1

1x

)dx 7ln25 1024[][][0.3][][][][](x2y2,Dy

x, y

xa, ya及y

D(a0[]3ayya ya

(x2

y2)dx 43a(2ay2a2y1a3)dy 7a 314a4 [][][0.3][][][]3x3x(2x0 0

y)dy.[]3(93x3x2)dx 50 2 227 102[][][0.3][][][][]D

1(xy

,D0

x1,0

y1.[]1x1 1 y 40 0xy)210

1 1x

12

)dx 7ln4 3[][][0.3][][][][]

,D:x y

2x,0

x1.D[]1x

2x

40 x11(2x2x2)dx 7021 106[][][0.3][][][][],Dyx, 1,x3D[]3xxy 41 1331(x31)dx 712 3101ln3 2[][][0.3][][][][](x2y2,Dy2,yx,

y2xD。[答案及評(píng)準(zhǔn)]2yy(x2

y2x)dx 402(19

y323y3

y2)dy 70136

24 810[][][0.3][][][]y[](x,D曲線x1 y

y1xy1D。[答案及評(píng)準(zhǔn)]1y0

y(x1)dx 41102

y(yy2)dy 7 1 1024[][][0.3][][][][]D

11x4

,Dyx

y0

x1。[答案及評(píng)準(zhǔn)]1 1

dxxdy 301x4 01 1 dxxdy 601x4 011x22 01x4 108[][][0.3][][][]4y2[],4y2

x0。D4y4y2

y2dy

42 02y2(4y2)dy 7064 1015],DD

y x2121

yx

4。]42

xx4y 41x21214218

(x2

4x x3)dx 7210]yy,Dyx,y0,x1。D]1exxxeyy 40 01ex(ex)x 70e2e1 2 2]D

,D曲1,y與x2y2。]解得交點(diǎn)(2,12

(2,4)原式2x2x21y 41 1 y2x2x2(x1234

1)dx 7x210[][][0.3][][][][](x2y2,D:1x2,0y1.D[]2x1(x21 02(x21)dx

y2)dy 471 322 103[][][0.3][][][]2[]4y2,Dx0,y ,y2

xD。[]24ydyyyxy)dx 420 024y)dy 72202 [][][0.3][][][][](xy2, D

D:0

y

x,0

x .2[]2dxx(x20 0

y2)dy 42(x13x)dx 720 37 109[][][0.3][][][]x[]x ,D拋物線y yxD。[](0,0)10

yxdx 4y2y11(y y4 y)y 7y2 06 1055[][][0.4][下二重積的二重積的][內(nèi)容]xdxdy,D2xD

y1

1x2,0x1.[]1xx0 21x(x0

x241x21)dx 71 6[][][0.3][極下二重積的二重積的][內(nèi)容]r2drdD

,其中

D:a

ra,0

(a0)2 .[]2da2

r2dr 40 a21a3)d723a2

0(2) 3 2 3[][][0.3][極下二重積的][][二重積的][內(nèi)容]利用極二重積x2y2

, 其中D:x2y

D1x0,y0.[]2d11r2)rdr 520 0 1r2)n1r2)(r21 84 0(2n2 4[][][0.3][下二次積分的二次積分的]二次積分

4x

x2y2[答案及評(píng)分準(zhǔn)]d2r20 0

2 05r[ ]2 r3 08 3[][][0.3][下二次積分的二次積分的]a2y2二次積分a2y20 0

(x2y2

(a0).[答案及評(píng)分準(zhǔn)]2dar30 02 a48[][][0.3][下二重積分的二重積分的]2d2er20 0(e4

x2y2

ex2y2dxdy .510[][][0.3][直角下二重積分的二重積分的][內(nèi)容]

, Dx2y

2,x

y2.D[答案及評(píng)分準(zhǔn)]11

dy2y2y2

xdx 41(2y2y4)y 7022 1015[][][0.3][][][][]e2ddy,Dyxy3D。[答案及評(píng)準(zhǔn)]1e2xxy 401(e0

x3x3ex2)dx 71 e1 12[][][0.3][][][][],D(x2)2y21上半圓x軸D。[答案及評(píng)準(zhǔn)]4xx234xx231 0

413x(4xx23)dx 7214 3[][][0.3][][][][],

D:x2

R2.D[答案及評(píng)準(zhǔn)]RR

y2dyR2R2y2R2y2

x3dx 4R2R2y2R2y2

x3dx被函數(shù)為奇函數(shù) 7故為. [][][0.3][][][]xy,D:2y2

a2,

y0.D[答案及評(píng)準(zhǔn)]2ax0

a2x242ax a2x2702a3 103[][][0.3][][][][]|x,D:D

a2 b2

1.[]D一象限部D上4倍在一象限1|xx,b4bdyb0 0

b2y2

42ba2

(b2y2)dy 70b24 a2b 103[][][0.4][][][][]x||y,

D:x2y2

1.D[]1e2xxy 501(e0

x3x3ex2)dx 81 e1 12[][][0.3][][][]x||y, Dx||y1.D[]41xx(x0 0541(11x2)x 80 2 24 103[][][0.4][][][][]

,D為y1,

y2xx0所圍成區(qū)x1yxD。[試題答案及評(píng)分標(biāo)準(zhǔn)]yyxyyx2 1 yyxdx5011yy101y11

yx2 y y

82 8 012

0 11

y19321 8 2[試題編號(hào)][計(jì)算題][較易0.4][直角坐標(biāo)系下二重積分的計(jì)算][][二重積分的計(jì)算][試題內(nèi)容]計(jì)算二重積分

x2y2D是以O(shè)(0,0)

為頂點(diǎn)D的三角形區(qū)。[試題答案及評(píng)分標(biāo)準(zhǔn)]1xx

x2y250 xy2x2yy2x2y2

x y

x dx01026

x2dx

arcsin )|2 x x810[試題編號(hào)][計(jì)算題][較易0.3][直角坐標(biāo)系下二重積分的計(jì)算][][二重積分的計(jì)算]sinx[試題內(nèi)容]分

,Dyx

y0

x1所圍成的區(qū)。xD[試題答案及評(píng)分標(biāo)準(zhǔn)]1

xdxxdy 40 x 01nx 701cos1 10[試題編號(hào)][計(jì)算題][較易0.3][直角坐標(biāo)系下二重積分的計(jì)算][][二重積分的計(jì)算][試題內(nèi)容]計(jì)算二重積分

sinx中 x

y所圍成的區(qū)。D[試題答案及評(píng)分標(biāo)準(zhǔn)]1sinxdxx2dy 40 x 01xnx 7010[][][0.3][下二重積分的二重積分的][內(nèi)容]x2y2,

D:x2y2

4

x0

y0.D[答案及評(píng)分準(zhǔn)]2d2r2)rdr 420 0574 1(5ln54) 104[][][0.3][下二重積分的二重積分的][內(nèi)容](x2y2

D:x2

2x,

x2y2

4x.D[答案及評(píng)分準(zhǔn)]222

d4cosr3dr 42cos2260cos4d72045 102[][][0.4][下二重積分的二重積分的][內(nèi)容]

x2

y2, D

2x.D[答案及評(píng)分準(zhǔn)]rdD222

d2cosr250282

d3 2216d823 0623 332 109[][][0.3][下二重積分的二重積分的][內(nèi)容]利用

x2

y2dxdy

Dx2y

4.D[答案及評(píng)分準(zhǔn)]2d2r2 60 016 103[][][0.3][下二重積分的二重積分的][內(nèi)容]利用

x2

y2dxdy,Dx2y

1.D[答案及評(píng)分準(zhǔn)]r22d1r20 0

65 6[][][0.3][下二重積分的二重積分的][內(nèi)容]

1x2y2dxdy

,Dx2y

1,

x0

y0.D[答案及評(píng)分準(zhǔn)]1r2rdrdD1r221r220 0

5 1 2 3 [ 2 2 36

r2)280[][][0.3][下二重積分的二重積分的][內(nèi)容]利用

ydxdy

, Dx2y

a2,

x0

y0D(a0).[答案及評(píng)分準(zhǔn)]rrdrdD2dar2dr 520 01a3 83a33[][][0.3][下二重積分的二重積分的]2[內(nèi)容]利用二重積分(x2y2)3dxdy2

,Dx2y

R2,

x0,Dy0,(R.[答案及評(píng)分準(zhǔn)]r3rdrd5D2dRr4dr 820 0R52 5R5 1010[][][0.3][下二重積分的二重積分的]內(nèi)容利用二重積分

x2

y2

, 其中

D:a2

x2y

b2,D(ba0).[答案及評(píng)分準(zhǔn)]r3D2dbr2 50

a1(b3a3)3 (b3a3) 3[][][0.4][下二重積分的二重積分的][內(nèi)容](4x2y2,

D:x2y

4.D[答案及評(píng)分準(zhǔn)]4 x2y24

dxdy

x2y24

(x2

y2)dxdy 22r3 50 016 8410[][][0.3][下二重積分的二重積分的][內(nèi)容]xydxdy

,其中

D:x2y

1,

x2y

2x

y0.D[答案及評(píng)分準(zhǔn)]3d2ar3cosdr 530 13(4cos51)d730 49 1016[][][0.4][下二重積分的二重積分的][內(nèi)容]利用xdxdy,D:x2y2

2x,x2y

x.D[答案及評(píng)分準(zhǔn)]rrdrdD222

d2cosr25cos212

(8cos3)d32214d823 07 8[][][0.4][下二重積分的二重積分的]

x2y2

,

D:x2y

R2

(R0),Dx0,y0.[答案及評(píng)分準(zhǔn)]r2D2dR1r2)rdr 520 01r2)ln(1r2)r2]R 82 2 0 R2)ln(1R2)R2] 104[][][0.4][下二重積分的二重積分的][內(nèi)容]利用二重積分sin x2y2dxdy,中D:1x2y

4

x0,Dy0.[答案及評(píng)分準(zhǔn)]sinrrdrdD2d2rsinrdr 520 1([rcosr]22cos82 1 1(cos12cos2sin2sin1) 2[][][0.3][下二重積分的二重積分的]

(63x2y

, 其中

D:x2y

R2,D(R0).[答案及評(píng)分準(zhǔn)]2dR6rs2rsn) 50 03R2R(6r2rsin)rdr 706R2[][][0.4][下二重積分的二重積分的][內(nèi)容]利用二重積分2x3y

,Dx2y

a2,

x0,Dy0, (a0).[答案及評(píng)分準(zhǔn)]2rcossin)rdrdDrdrdr2(2cos3sin)drDD D2a2(2cos3sin)ar224 0 0a3a2(2a34 3(5a)a24 3[][][0.3][下二次積分的二次積分的]二次積分3x2

y2dy.1[答案及評(píng)分準(zhǔn)]

x12ny2yyx 40 12ysiny2701cos4) 102[][][0.3][下二次積分的二次積分的]二次積分1xxx0 0

1x2y2.[答案及評(píng)分準(zhǔn)]1y1

1x2y240 y11(y3)dy 7301 104[][][0.3][下二次積分的][][二次積分的]二次積分1x2x1ey2y.0 x[答案及評(píng)分準(zhǔn)]1ey2yyx2x 40 011y3ey2y 73011 106 [][][0.3][下二次積分的二次積分的]1y3二次積分11y3

xy 0 x2[]1y31y3

y dy

5y0 0y1y311 y2 y1y32 021(23

[][][0.3][下二次積的二次積的]1x1x

y3dy.0 x[]1yyx0 0

y3dx 510

y3121

y2dy 816[][][0.3][下二次積的][二次積的]1y1n2x.0 y[]1xxnx2y 50 01xnx2x 8012[][][0.3][下二次積的二次積的][內(nèi)容]

y

sinxdx.2 20 y 2 2[]22

xdxxdy 5001

x 028210[][][0.4][][][][]D

,Dy1x

y2

x1x2。[答案及評(píng)準(zhǔn)]1y2eyx2y2eyx 51 1 1 12 y1(e2ye)y2(e2yey)y 81 12e2(e22

[][][0.4][][][][]|y2x,

D:0x1,0

y2.D[答案及評(píng)準(zhǔn)]1x2x(2x)x1x20 0 0 21(424x)x0

(y2x)dy 584 3[][][0.4][][][][]|yx,

D:0x1,0

y1.D[答案及評(píng)準(zhǔn)]1xx(x)x1yy(y)x 50 0 0 021xx(x)y 80 01 3[][][0.4][極][][]1x2y2[]D

1x2y2

, D:

1.[答案及評(píng)準(zhǔn)]2d11r20 01r11udu01u

58(2ln2[][][0.4][下二重積分的二重積分的][內(nèi)容](x2

y2

a2x2y2dx

,Dx2y

a2,Da0.[答案及評(píng)分準(zhǔn)]2dar0 0

a2r25令rat2a53tsin5t)dt 8204a5 1015[][][中等0.5][下二次積分的][][二次積分的]4x2二次積分14x20 1x2

ex2y2

2x44x2

ex2y2dy.[答案及評(píng)分準(zhǔn)]2d2er2 520 1 (e4e) 4[][][中等0.5][下二次積分的][][二次積分的]R[內(nèi)容]積分R

2x(x2y2

(x2y2)dy

(R0).R2R2x2R2[答案及評(píng)分準(zhǔn)R24dRr3 640 0 R4 1016[][][中等0.5][下二次積分的][][二次積分的][內(nèi)容]

R2ey2yex2

ey2

R2x2ex2

(R0).0 0 R 02[答案及評(píng)分準(zhǔn)]2dRer2 62 04eR2) 8[][][中等0.5][下二重積分的][][二重積分的][]D

1x2y1x2y

dxdy

D:x2y

a2(0

a.[]1r42darr31r40 01r41r4

a r3

] 70 01r4[arcsinr2]1r40

[a 1r42 1a4a21a4[][][0.5][下的][][的]a2x2aa2x2

1 , (a0)4a24a2x2y2

0 x04

d2asin0

r 54a2r24a2r24

4a2r

asind702a04

cos)da(2

2 2) 10[][][0.5][下的][][的][](4x

,

D:x2y

2y.D[]設(shè)xr, yrsind2sin(4rsrn) 50 08n21n3s1n4)d 80 3 310[][][0.5][下次的次的]2xxnxy4nxy.1 x 2y 2 2y[]2dyy2sinxdx 41 y 2y22y(cosy)dy 71 2 42) 2 [][][0.6][][][]1 x

y2[][]

dx 0 0

2dy.0

y1y2

y22dx 410

22y2)dy

y2

1

y2e 2dy ye 2dy 70 0ee

y2

1

y2 )2dy 2e22dy 2e2dy2e1 102[][][0.7][][,][]

lim

x2y2.[]

t0t2x2y21m1(r2) 50 0 m1n0 2(unuu)1 80 t2[][][0.5][][][][],D(x1)2y21(y1)2

1D[]xryrsin4 2sinr2sr2cosr2s 54 0 0 0488sns s83 44 2 703 34 1 4 2[][][0.5][下二重積的二重積的][內(nèi)容](x

y

, Dx2y2xy.D[]sin)D43sin)dsinr254 0441sin)4d4344 3 3 4

sin4

)d844sin4tdt3 028sin4tdt230 102[][][0.5][下二重積的二重積的]D

1x2y

, Dx2y24

x2y

16,x2y24x.[]3(430 4cos

r4r) 5 232cos)d223 70 334 1033[][][0.5][下二重積分的二重積分的][內(nèi)容]

x2y2dxdy

,Dx2y

4

x2y

2x.D[答案及評(píng)分準(zhǔn)]22420 2cos

r2r22r2 5 022220

8 (1cos3 )d 7 3 33 9[][][0.6][下二重積分的二重積分的][內(nèi)容][答案及評(píng)分準(zhǔn)]

x2y2

|x2y24.當(dāng)4x2y

9時(shí)

|x2y24x2y24,x2y

4時(shí)

|x2y244x2y2.原式

4x2y2

(x2y2

x2y2

(4x2y2523(r2)r22r(4r2) 80 2 0 02(81294884)441 102[][][0.8][][][,]] ft)Fu) x2y2z2(2u)2

ef

x2y2z2,F(u。[答案及評(píng)標(biāo)準(zhǔn)]14x2Fu)x y2ur2nef(r)14x20 0 02r2ef(r)60F(u)32u2ef(2u) [][][0.7][][ ][][]

,z3x2

y2z1x2y0部分立體。[答案及評(píng)標(biāo)準(zhǔn)]2V2

dx

yx2

414x21 0 3x214x2214x2214x22

y(14x2y2)dy1 0222

4x2)28114122 1015[][][0.65][][ ]xy[]xy

x2

,其中是由曲面z ,x

y10z0所圍界閉區(qū)域。[答案及評(píng)標(biāo)準(zhǔn)]xyV1xxy x2z 4xy0 0 01xx1x3 60 0 211x3104dx 8 1 10240[][][0.65][][ ][]

yx,y

x, y0

z0

xz2。[答案及評(píng)標(biāo)準(zhǔn)]Vxxyxy(xz)z2 2 40 0 02dx20 0

x)dy 6220

xx)dx 821 1016 2[][][0.7][][ ][][]z0，(a0).

xy為：x

yza，x0，y0，[答案及評(píng)標(biāo)準(zhǔn)]Vaxaxyaxy(xy)z 40 0 0axax(xy)sa 60 0aax(ax)cos80aacosa1a22

cosa [][][0.7][][ ][]]1z1x1

y2dy.0 0 x[答案及評(píng)標(biāo)準(zhǔn)]I1yyx10 0 0

y2dz 51y0

y2812[][][0.6][][ ][][]

(x

dvy

，:1

x2，1

y2,1

z2.[]I2x2y1 1 112x2[

(x1

1yz)3dz 1

2521

(xy1)2

(x

y2)212[ 1

2

1 821 x

x

x27235 2 2[][][0.6][][ ][][]1

x3y2

x1

x2,

y0

yx2,z0,z x[]xI2xx2y1x3y2z 4x1 0 02xx2y2 61 012x2 861255 1048[][][0.65][][ ][]

z

xy

xy1,

z0所。[]I1xxyyz 40 0 021x2(1x)3dx 713 1 10180[ ]。

xzy2

dvx0,

z0,

z1

y2x

y答案評(píng)標(biāo)準(zhǔn)]I1y

yxy2

dz 40 0 0

y2)1y

yx1

y2)260 011y1

2(1y2)2dx 84 01 1048[ ]設(shè)是x1,y。答案評(píng)標(biāo)準(zhǔn)]

xy2xz0z

5x2

y2有界閉區(qū)域。5x2y2I1x5x2y2

40 x 01x2

x(5

x2

y2)dx 60 x 21(5x25x4)dx 80 2 31 102[ ][],yz,

z0

yx0x

。[答案及評(píng)標(biāo)準(zhǔn)]Ixnxyyz 40 0 0xnx 60 008

1x2

810[編號(hào)][][難0.65][三重積][ 三重積][]2

,x2

1,

y0

y1位z0立體。[答案及評(píng)標(biāo)準(zhǔn)]1x221x1y 21x20 0 021x1y1x2) 60 011x2) 802 3[編號(hào)][][難0.7][三重積][ ][三重積][]7xy2z3dv,yx,

x1, z

xyzx2y有界閉。[答案及評(píng)標(biāo)準(zhǔn)]1xxy

7xy2z3dz 40 0 x2y1xx7x51x4)y6 60 0411xx21x4) 84 0 0 1 10221[][][0.6][][ ][]]ex2y2dv,0[答案及評(píng)標(biāo)準(zhǔn)]1x1yy3ex2y2z0 0 01x1ex2y3ey20 0

x1,0

y10zxy3。35e1 4[][][0.6][次][ ][次]xxyy()3z.0 0 0[答案及評(píng)標(biāo)準(zhǔn)]zy()3 50 z y()3zy)0 z1z)2()3z 72 01cos3) 106[][][0.6][次][ ][次]1x1y1

1z4.0 x y[答案及評(píng)標(biāo)準(zhǔn)]1zzyy0 0 0

1z451zzy0 0

1z4dy1103

1z4dz 721(2218

10[][][0.6][][ ][][]

1dv.x1x2z0xx2y2

yy

z。[答案評(píng)分標(biāo)準(zhǔn)]2xxy1 0 0

1x2y2dz 42xx y 61 0x2y2821n(x2y2)x812021ln2dx 91212 2[編號(hào)][][難0.65][三重積分的][ 三重積分的][] 是x

yz

算

1.(xyz1)3.[答案評(píng)分標(biāo)準(zhǔn)]1xxyxy 1 z 30 0

(1x

yz)31xx1 1

15 0 0 2xy)2 411 1 x3 82 1x 4 0 125) 2 8[編號(hào)][][難0.65][三重積分的][ 三重積分的][]

,是0

x1,0

y1,0z

x

y確定的立體。[答案評(píng)分標(biāo)準(zhǔn)]1y1xx1y)z 20 0 01y1[x1y0 01[1yy1y 6021 102sin2xy2[][][0.65][sin2xy2[0x,。[答案及評(píng)標(biāo)準(zhǔn)]

yx,

0z