第十章重积分试题库(无水印)

、 知识点(第十章重积分)0101二重积分的概念010101二重积分的定义010102010103二重积分的物理意义0102重积分的性质010201二重积分的性质010202二重积分的对称性0103二重积分的直角坐标计算法010301用直角坐标计算二重积分010302交换积分次序0104二重积分的极坐标计算法010401二重积分化为极坐标系下二次积分三重积分0201三重积分的概念020101三重积分的概念020102三重积分的性质020103三重积分的对称性0202三重积分的计算用直角坐标计算三重积分用柱面坐标计算三重积分用球面坐标计算三重积分0301几何应用030101计算曲面的面积030102计算立体体积0302物理应用030201计算物体质心030202计算转动惯量030203计算引力[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),

选),m

f(,)i i

0

i1

i i i(i。

i,n) []fxyD上[100103][][0.2][何知识2][][]D面S= .D[]2S.[100104][][较0.3][何含][][何含,坐系下][]D面S,坐系下Drdrd= .D[]S.[100105][][较0.3][何][角坐系下计算][计算][]D(0,0)))三角形区域计算二重积分xy

=___________.D[答案评分标准]16.[100106][填空][易0.2][二重积分物理意义][二重积分计算][二重积分计算,二重积分物理意义][]D其上(xy处面密度(xy如果(xy)在D上连续则薄片质量m= _.[答案评分标准](xy)d ((x).D D[100107][填空][较易0.3][二重积分几何意义][二重积分计算][二重积分几何意义,二重积分计算]根据二重积分

1x2

y2d

=___________ 其中D：x2y2

D1.[答案评分标准]2.3[100108][填空][较易0.3][二重积分几何意义二重积分概念][]

f(t连续函数z0x2y2

1zf(xy)]2所围立体体积可用二重积分表示_.[答案评分标准]x2y2

[f(xy)]2dxdy[100109][填空][较易0.4][二重积分几何意义][][二重积分,极坐标系下二重积分计算][]

D:0r1,02

，1r2rdrd= _.D[答案评分标准]16][][]

fx,yDf(x,y)0,

f(x,y)d

D[]z

f(x,y)D。][0.3][计算][][]D:0y

a2x2,0x

知 a

x2

y2dxdyD= .[]1a36][较0.3][][计算][][]Dx2y2

2,由

2x2

y2dxdy= .D[]4 234 23][较0.4][][计算][][]Dx2y2

2x由

2xx2

y2dxdy= .D[]2.3][较0.3][对称性][计算对称性][]Dx2y2

4y0,x3y2

= .D[0][较0.3][对称性][计算对称性][]

fx,yy轴对称D

f(x,y)f(x,y)，则f(x,

=__________.D[0][0.3][][][]D:x2y2a2,y0m为奇数时， xmyn= _.D[0][0.3][][][]D0xaa

ya,n

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

f(x交换次序后为_______________.[]1x0 0

f(x,y)dy.][0.4][交换][][交换次序]f(x,y)1yy0 y

f(x,

交换次序后为______________.[]1x

f(x,y)dy0 x2[100120][][0.4][交换][][交换次序][]f(x,y)为连续函数，次axx0 0

f(x,y)dy

交换次序后为_________________.[]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

在下先对r为 .[]

1 00 0

f(rcos,r)rdr[100124][][0.5][下][][下]aa

dx

a2x2a2x

f(x,y

在下先对r .[]20 0

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

x

f(x,y

在下先对r为 .[]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上连续Dx2y2

4

f(x2

y2

Dr.[]2d0 0

f(r2[][][0.3][几何意义][][计算,几何意义][]

4x2

y2dxdy

= D:x2y

Dy0.[]4.3[][][0.3][几何意义][][计算,几何意义][]根据几何意义 a

x2

y2dxdy =

Dx2y

a2

y

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

x

0

xxD

y2

= .[]13[][][0.4][][][,][]。则质量公式为 .[]M=[][][中等0.6][][直线][,][]D0xx yy。则关于直线 0

0z

为 .[]

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

B2

C2[][][0.35][对称性][][]

(ey2|y|1

sin

y3z2x3)dv

则I= 。[]I24[][][0.35][对称性][][]

x2y2z1

[x3ezx2)

= 。[答案及评分标准]I4[编号填空易0.2][重积分的性质三重积分][]

f(x,y,z)在有界闭区域上可积， ，则2I

f(x,y,z)dv

f(x,yz)dv。 1[答案及评分标准]2 1

f(x,y,z)dv[编号填空易0.2][重积分的性质三重积分][]

(x23y2

(3xy1x2)dv2 [答案及评分标准]I(3xy3x23y2)dv2[编号填空][0.4][重积分的物理应用三重积分]设(x,y,z)设M(x,y,z)为其重心，关于xoy平面的静矩定义为：Mxy .[答案及评分标准]

,

M 的三重积分计算式为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则

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 。[答案及评标准]B[][][0.2][][][][]f(x,y)Df(x,y)d

DA ； B C D 。[答案及评标准]C[][][0.3][直角坐标系下计算][]xydx

(D0

x2,0

1)值为D1 1 1 1A B C D6 12 2 4[答案及评标准]B计算][对称性][计算][]若区D为0

x2,|

2,则xy2dx=A 0 B 323

DC 64 D 3[答案及评标准]A计算][性质][计算][]设

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次序结果为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][下二重积的计算二重积][内容]

f(x,y)1y0

3yy22

f(xy)dx可交换积次序为1x

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][下二重积的计算二重积][内容]

fxy为连续函数，则积分1xx2

f(x,y)y2x2

f(x,y)dy0 0 1 0可交换积次序为1y

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[][][较易0.4][极下二重积的计算二重积的计算][]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上连续函数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上连续函数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有界函数。若m

n

f i j k 13n

i1

jk

( , , )( nn n n

I则A f(x,y,z)积 B f(x,y,z)一定

f(xyzI0

f(xyz必试题答案及评分标准]B]试题内容F(xyz有界闭域(xy

f(x,y,

(x,y,z)，则：1 2F(x,y,f(x,y,f(x,y,1 2 A 式成立 B 式成立C f(x,y,时成立 D1

f(xy也未必成立1试题答案及评分标准]C]试题内容设,是空间有界闭区域，

f(xyz, , 2

f(x,y,

f(x,y,

f(x,y,z)dv的充要条件是 3 1 2A f(x,y,z)是奇函数 B4

f(x,y,z)0

(x,y,4C 4

D

f(x,y,z)dv0试题答案及评分标准]D

4答( )]试题内容设f(xyz是一全空间的连续函数，由中值定理

f(x,y,

f(,,V.(,,而V为的体，则：A f(xyzxyz为奇函数时B f(,,)0

f(,,)0若x2y2

1

f(,,)

f(0,0,0)f(,,xyz的奇偶性无必然联系试题答案及评分标准]D[][][0.3][三重积分的性质三重积分的性质]

( )[内容]设uf(t在(是上半单位x2y2

1,z0,

I

f(xy,则A I0

B I0I0

I的符不定[案及评分标准]B[][][0.3][三重积分的性质三重积分的性质][内容]设u

ft)是(,)|x1,|y1,|z1

I

f

a,b,c为常数，则I0

I0I0 D I的符由ac确定[案及评分标准]C[][][0.3][三重积分的性质三重积分的性质]

( )[内容]设uf(t是(上严格单调减少的奇函数，I x2y2z2A I0

kf(xyz

k0B

I0C I0 D 当k0I0;当k0I0[案及评分标准]A[][][0.3][三重积分的性质三重积分的性质]

( )内容为单位球体

x2y2

1

位于1

z

部分的半球体，I(xyf(x2y2z2,则I0

I0I0

D I(xyf(x2y2z21[案及评分标准]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.则A dvx99dv

. B y99dvdv Cx9v4y9C

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

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

y0

z02xyz1

则

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确定立体体A 1r

1r2dz

B rr

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

0 0 r2[][][0.5][三重积化为三次积三重积的计算][内容]设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][三重积化为三次积x2y2[内容]设是由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。[答案及评准]D

f(x,y)dx

1xx1y2y1 50 0 6而D当面1, 71f(x,y)D.6 10[][][0.2][次][][次]3y2(x2)x1 1[答案及评准]1原式(31)(3

x3x)2 712(78 3 3[][][0.2][次][][次]4x2

ydy.[答案及评准]43xdx

2 x x522=9. 10[][][0.2][次][][次]2ynyexx.1 0[答案及评准]21

1)dy1 102[][][0.2][次][][次]2 2 []x1 0

xydy.[答案及评准]原式

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

xydy.[]9

xdx4

51 0832= . 109[][][0.2][下二次积的二次积的][内容]

x

y4dy20 cosx[]220

151dx 5=8. 1010 75[][][0.3][下二次积的][][二次积的]xsxy2ny0 0[]1n1s)3x 5304= . 103[][][0.3下二重积的二重积的][内容]

ysyx2n

ydx2 02[]2922

sin

ycos3

5=12. 105[][][0.3][下二重积分的][][二重积分的][内容]D

11y2

d,Dx2,|y1.[答案及评分准]2x1 1 y 52 11y2=42arctan1.. 10[][][0.2][下二重积分的二重积分的][内容]

D:0

x1,0

y2.D[答案及评分准]1x2y 40 01x2y. 70 0=1 10[][][0.2][下二重积分的二重积分的][内容]

,

D:0

xa,0

yb.D[答案及评分准]=a

dxb

50 024(ab)3 29[][][0.2][下二重积分的二重积分的][内容]D

y d,D01x

x1,0

y2.[答案及评分准]1 1 x2y 501x 02ln2 [][][0.2][下二重积分的二重积分的][内容]ex,

D:0

x1,0

y1.D[答案及评分准]1exx1eyy 50 0(e[][][0.2][下二重积分的二重积分的][内容]D

x21y2

,D0

x1,0

y1.[答案及评分准]1x2x1 1 y 50 01y218 1012[][][难程度][1][关键词][][][0.2][下二重积分的][][二重积分的][答案及评分准]1x2x2y 51 04 3[][][0.2][下二重积分的二重积分的][内容]D

x 1y2

, D0

x1

y1.[答案及评分准]2x1 1 y 40 11y222arctan1 7 10[][][0.2][下二重积分的二重积分的][内容]D

,

D:0

x,0

y .2[答案及评分准]2sinxsy 520 02 [][][0.2][下二重积分的二重积分的][内容]分,中D:1

x3,0

y2.y1D[答案及评分准]式3x2x2 1 y 51 01y28ln3 103[][][0.2][下二重积分的二重积分的][内容]

,D0x1,0

y4.D[答案及评分准]式13x4y 50 03 10[][][0.2][下二重积分的二重积分的][内容]xsinD

D:1

x2,0y2.[答案及评分准]2原式2dx21 0

53 102[][][0.3][下二重积分的二重积分的][内容](x

d

,

D,|y1.D[答案及评分准]原式0

x1

y1

yx 510[][][0.3][下二重积分的二重积分的][内容]x(x,

D3,|y1.D[答案及评分准]原式3

x2x1

y3

x1

510[][][0.2][下二重积分的二重积分的][内容](x3

,

D1,0y1.D[答案及评分准]

3x1y

x1y2y1 023

1 0[][][0.2][][][][],DO(0,0),D。[答案及评准]1x1y 40 x11)x 701 6[][][0.2][][][][],

D:0x1

y0.D[答案及评准]1x

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

D:0

yx,0x.D[答案及评准]xnx(xy2)y 40 0(xinx1in3)dx 70 34 109[][][0.3][][][][],D由曲线y,线y0, x2,所围成D。[答案及评准]2x0 0

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

,Dyx,

y2x

x4。D[答案及评准]原式4x2x

xydy 4043x2

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

D:xy

x2.D[答案及评准]原式2x

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

x。原式x(x

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

yx1,

y1,Dy3。[答案及评准]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。[答案评准]1x5x(x6)y 40 x16x2x 70251 103[][][0.3][][][][],D

y1

yxx2D。[答案评准]2xxy

x4121

1xx(x2

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

ydxdy,Dyx

xx2x4。[答案评准]41dx2xdy 42x x437229 [][][0.3][][][]yy,Dxy1.D[答案评准]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。[答案及评准]2yy(x2

y2x)dx 402(19

y323y3

y2)dy 70136

24 810[][][0.3][][][]y[](x,D曲线x1 y

y1xy1D。[答案及评准]1y0

y(x1)dx 41102

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

11x4

,Dyx

y0

x1。[答案及评准]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与x2y2。]解得交点(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抛物线y yxD。[](0,0)10

yxdx 4y2y11(y y4 y)y 7y2 06 1055[][][0.4][下二重积的二重积的][内容]xdxdy,D2xD

y1

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

x241x21)dx 71 6[][][0.3][极下二重积的二重积的][内容]r2drdD

,其中

D:a

ra,0

(a0)2 .[]2da2

r2dr 40 a21a3)d723a2

0(2) 3 2 3[][][0.3][极下二重积的][][二重积的][内容]利用极二重积x2y2

, 其中D:x2y

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

4x

x2y2[答案及评分准]d2r20 0

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

(x2y2

(a0).[答案及评分准]2dar30 02 a48[][][0.3][下二重积分的二重积分的]2d2er20 0(e4

x2y2

ex2y2dxdy .510[][][0.3][直角下二重积分的二重积分的][内容]

, Dx2y

2,x

y2.D[答案及评分准]11

dy2y2y2

xdx 41(2y2y4)y 7022 1015[][][0.3][][][][]e2ddy,Dyxy3D。[答案及评准]1e2xxy 401(e0

x3x3ex2)dx 71 e1 12[][][0.3][][][][],D(x2)2y21上半圆x轴D。[答案及评准]4xx234xx231 0

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

D:x2

R2.D[答案及评准]RR

y2dyR2R2y2R2y2

x3dx 4R2R2y2R2y2

x3dx被函数为奇函数 7故为. [][][0.3][][][]xy,D:2y2

a2,

y0.D[答案及评准]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为y1,

y2xx0所围成区x1yxD。[试题答案及评分标准]yyxyyx2 1 yyxdx5011yy101y11

yx2 y y

82 8 012

0 11

y19321 8 2[试题编号][计算题][较易0.4][直角坐标系下二重积分的计算][][二重积分的计算][试题内容]计算二重积分

x2y2D是以O(0,0)

为顶点D的三角形区。[试题答案及评分标准]1xx

x2y250 xy2x2yy2x2y2

x y

x dx01026

x2dx

arcsin )|2 x x810[试题编号][计算题][较易0.3][直角坐标系下二重积分的计算][][二重积分的计算]sinx[试题内容]分

,Dyx

y0

x1所围成的区。xD[试题答案及评分标准]1

xdxxdy 40 x 01nx 701cos1 10[试题编号][计算题][较易0.3][直角坐标系下二重积分的计算][][二重积分的计算][试题内容]计算二重积分

sinx中 x

y所围成的区。D[试题答案及评分标准]1sinxdxx2dy 40 x 01xnx 7010[][][0.3][下二重积分的二重积分的][内容]x2y2,

D:x2y2

4

x0

y0.D[答案及评分准]2d2r2)rdr 420 0574 1(5ln54) 104[][][0.3][下二重积分的二重积分的][内容](x2y2

D:x2

2x,

x2y2

4x.D[答案及评分准]222

d4cosr3dr 42cos2260cos4d72045 102[][][0.4][下二重积分的二重积分的][内容]

x2

y2, D

2x.D[答案及评分准]rdD222

d2cosr250282

d3 2216d823 0623 332 109[][][0.3][下二重积分的二重积分的][内容]利用

x2

y2dxdy

Dx2y

4.D[答案及评分准]2d2r2 60 016 103[][][0.3][下二重积分的二重积分的][内容]利用

x2

y2dxdy,Dx2y

1.D[答案及评分准]r22d1r20 0

65 6[][][0.3][下二重积分的二重积分的][内容]

1x2y2dxdy

,Dx2y

1,

x0

y0.D[答案及评分准]1r2rdrdD1r221r220 0

5 1 2 3 [ 2 2 36

r2)280[][][0.3][下二重积分的二重积分的][内容]利用

ydxdy

, Dx2y

a2,

x0

y0D(a0).[答案及评分准]rrdrdD2dar2dr 520 01a3 83a33[][][0.3][下二重积分的二重积分的]2[内容]利用二重积分(x2y2)3dxdy2

,Dx2y

R2,

x0,Dy0,(R.[答案及评分准]r3rdrd5D2dRr4dr 820 0R52 5R5 1010[][][0.3][下二重积分的二重积分的]内容利用二重积分

x2

y2

, 其中

D:a2

x2y

b2,D(ba0).[答案及评分准]r3D2dbr2 50

a1(b3a3)3 (b3a3) 3[][][0.4][下二重积分的二重积分的][内容](4x2y2,

D:x2y

4.D[答案及评分准]4 x2y24

dxdy

x2y24

(x2

y2)dxdy 22r3 50 016 8410[][][0.3][下二重积分的二重积分的][内容]xydxdy

,其中

D:x2y

1,

x2y

2x

y0.D[答案及评分准]3d2ar3cosdr 530 13(4cos51)d730 49 1016[][][0.4][下二重积分的二重积分的][内容]利用xdxdy,D:x2y2

2x,x2y

x.D[答案及评分准]rrdrdD222

d2cosr25cos212

(8cos3)d32214d823 07 8[][][0.4][下二重积分的二重积分的]

x2y2

,

D:x2y

R2

(R0),Dx0,y0.[答案及评分准]r2D2dR1r2)rdr 520 01r2)ln(1r2)r2]R 82 2 0 R2)ln(1R2)R2] 104[][][0.4][下二重积分的二重积分的][内容]利用二重积分sin x2y2dxdy,中D:1x2y

4

x0,Dy0.[答案及评分准]sinrrdrdD2d2rsinrdr 520 1([rcosr]22cos82 1 1(cos12cos2sin2sin1) 2[][][0.3][下二重积分的二重积分的]

(63x2y

, 其中

D:x2y

R2,D(R0).[答案及评分准]2dR6rs2rsn) 50 03R2R(6r2rsin)rdr 706R2[][][0.4][下二重积分的二重积分的][内容]利用二重积分2x3y

,Dx2y

a2,

x0,Dy0, (a0).[答案及评分准]2rcossin)rdrdDrdrdr2(2cos3sin)drDD D2a2(2cos3sin)ar224 0 0a3a2(2a34 3(5a)a24 3[][][0.3][下二次积分的二次积分的]二次积分3x2

y2dy.1[答案及评分准]

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

1x2y2.[答案及评分准]1y1

1x2y240 y11(y3)dy 7301 104[][][0.3][下二次积分的][][二次积分的]二次积分1x2x1ey2y.0 x[答案及评分准]1ey2yyx2x 40 011y3ey2y 73011 106 [][][0.3][下二次积分的二次积分的]1y3二次积分11y3

xy 0 x2[]1y31y3

y dy

5y0 0y1y311 y2 y1y32 021(23

[][][0.3][下二次积的二次积的]1x1x

y3dy.0 x[]1yyx0 0

y3dx 510

y3121

y2dy 816[][][0.3][下二次积的][二次积的]1y1n2x.0 y[]1xxnx2y 50 01xnx2x 8012[][][0.3][下二次积的二次积的][内容]

y

sinxdx.2 20 y 2 2[]22

xdxxdy 5001

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

,Dy1x

y2

x1x2。[答案及评准]1y2eyx2y2eyx 51 1 1 12 y1(e2ye)y2(e2yey)y 81 12e2(e22

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

D:0x1,0

y2.D[答案及评准]1x2x(2x)x1x20 0 0 21(424x)x0

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

D:0x1,0

y1.D[答案及评准]1xx(x)x1yy(y)x 50 0 0 021xx(x)y 80 01 3[][][0.4][极][][]1x2y2[]D

1x2y2

, D:

1.[答案及评准]2d11r20 01r11udu01u

58(2ln2[][][0.4][下二重积分的二重积分的][内容](x2

y2

a2x2y2dx

,Dx2y

a2,Da0.[答案及评分准]2dar0 0

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

ex2y2

2x44x2

ex2y2dy.[答案及评分准]2d2er2 520 1 (e4e) 4[][][中等0.5][下二次积分的][][二次积分的]R[内容]积分R

2x(x2y2

(x2y2)dy

(R0).R2R2x2R2[答案及评分准R24dRr3 640 0 R4 1016[][][中等0.5][下二次积分的][][二次积分的][内容]

R2ey2yex2

ey2

R2x2ex2

(R0).0 0 R 02[答案及评分准]2dRer2 62 04eR2) 8[][][中等0.5][下二重积分的][][二重积分的][]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[]设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][下二重积的二重积的][内容](x

y

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

sin4

)d844sin4tdt3 028sin4tdt230 102[][][0.5][下二重积的二重积的]D

1x2y

, Dx2y24

x2y

16,x2y24x.[]3(430 4cos

r4r) 5 232cos)d223 70 334 1033[][][0.5][下二重积分的二重积分的][内容]

x2y2dxdy

,Dx2y

4

x2y

2x.D[答案及评分准]22420 2cos

r2r22r2 5 022220

8 (1cos3 )d 7 3 33 9[][][0.6][下二重积分的二重积分的][内容][答案及评分准]

x2y2

|x2y24.当4x2y

9时

|x2y24x2y24,x2y

4时

|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。[答案及评标准]14x2Fu)x y2ur2nef(r)14x20 0 02r2ef(r)60F(u)32u2ef(2u) [][][0.7][][ ][][]

,z3x2

y2z1x2y0部分立体。[答案及评标准]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所围界闭区域。[答案及评标准]xyV1xxy x2z 4xy0 0 01xx1x3 60 0 211x3104dx 8 1 10240[][][0.65][][ ][]

yx,y

x, y0

z0

xz2。[答案及评标准]Vxxyxy(xz)z2 2 40 0 02dx20 0

x)dy 6220

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

xy为：x

yza，x0，y0，[答案及评标准]Vaxaxyaxy(xy)z 40 0 0axax(xy)sa 60 0aax(ax)cos80aacosa1a22

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

y2dy.0 0 x[答案及评标准]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答案评标准]I1y

yxy2

dz 40 0 0

y2)1y

yx1

y2)260 011y1

2(1y2)2dx 84 01 1048[ ]设是x1,y。答案评标准]

xy2xz0z

5x2

y2有界闭区域。5x2y2I1x5x2y2

40 x 01x2

x(5

x2

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

z0

yx0x

。[答案及评标准]Ixnxyyz 40 0 0xnx 60 008

1x2

810[编号][][难0.65][三重积][ 三重积][]2

,x2

1,

y0

y1位z0立体。[答案及评标准]1x221x1y 21x20 0 021x1y1x2) 60 011x2) 802 3[编号][][难0.7][三重积][ ][三重积][]7xy2z3dv,yx,

x1, z

xyzx2y有界闭。[答案及评标准]1xxy

7xy2z3dz 40 0 x2y1xx7x51x4)y6 60 0411xx21x4) 84 0 0 1 10221[][][0.6][][ ][]]ex2y2dv,0[答案及评标准]1x1yy3ex2y2z0 0 01x1ex2y3ey20 0

x1,0

y10zxy3。35e1 4[][][0.6][次][ ][次]xxyy()3z.0 0 0[答案及评标准]zy()3 50 z y()3zy)0 z1z)2()3z 72 01cos3) 106[][][0.6][次][ ][次]1x1y1

1z4.0 x y[答案及评标准]1zzyy0 0 0

1z451zzy0 0

1z4dy1103

1z4dz 721(2218

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

1dv.x1x2z0xx2y2

yy

z。[答案评分标准]2xxy1 0 0

1x2y2dz 42xx y 61 0x2y2821n(x2y2)x812021ln2dx 91212 2[编号][][难0.65][三重积分的][ 三重积分的][] 是x

yz

算

1.(xyz1)3.[答案评分标准]1xxyxy 1 z 30 0

(1x

yz)31xx1 1

15 0 0 2xy)2 411 1 x3 82 1x 4 0 125) 2 8[编号][][难0.65][三重积分的][ 三重积分的][]

,是0

x1,0

y1,0z

x

y确定的立体。[答案评分标准]1y1xx1y)z 20 0 01y1[x1y0 01[1yy1y 6021 102sin2xy2[][][0.65][sin2xy2[0x,。[答案及评标准]

yx,

0z