北大版高数第八章习题解答

北大版高等数学习题解答

8.1

LO(0,0,0),A(0,0,1),B(0,1,1),C(1,1,1)1.L(xy+yz+zx)ds,

.

1 1 1

.I=00dz+0ydy+0(x+1+x)dx=5

4.

2)2asin

1

t

15

a3.

L

5.

Cx′2+y′2=

.

a2+b2t2

√

C

a2+b2

a

.

.

4.

C(x

+y)ds,

a

2

:x=a

√

4

r2=a2cos2θ

πa2

cos2θsinθ, π

√a

cos2θ(cosθ+sinθ)cos2θdθ=+

y2

2.ab(a b3)

3

I=

x2+y2ds,

(0≤t≤2π).

π

a2sin2θ+b2cos2θdθ=

L

x=a(cost+tsint),y=a(sint tcost),

2π

.x2+y2=a2(1+t2),x′2+y′2=a2t2,I=

y z5)ds,

2

a2

√,L1

2

1

9.

b<a).

y=x,

0≤x≤1;L2:

z=0,

1√.I=02x

x2

LL1,L2x=1,

0≤z≤1.

y=1,

L2

L1:

5 1

6

=

√5

6

.

b2

=1

(x,y) π

|y|,

(0<

.

2a2ba2 b2

arcsin

√

L,m=

a

.

|y|ds=4L

a2sin2θ+b2cos2θdθ=2b2+

1.

(1)

8.2

2xydx x2dyL

,

L

O(0,0)

A(2,1):

3.

1

北大版高等数学习题解答

(2)I=

1

0dx 04dy= 4. 1 2

(4)I= 00dy+02xdx=4.

2.Fdr,F=(x2+y,x+y2),LL√(1)y= AB;(3)(0, 1).(3)I=

2

0 20

2xx

2

2

=0.

A(1,0)B( 1,0):ACB,C

.(1)L:x=cost,y=sint,t

2

(cost+sint)costdt= 2

π.I=

π

(cos2t+sint)( sintdt)+

3.

0 1

(3)I=1(x2+x 1)dx+(x+(x 1)2)dx+0(x2 x 1)dx+(x+( x 1)2)( dx)= 2

a2

+

y2

(2)I=

1

9

.

1

(x2 2x)dx=

2

7.

6

dx+dyL

|x|+|y|,

1

.

9.

35.

L(z2

. =3

π

y2)dx+(x2 z2)dy+(y2 x2)dz,Lx2+y2+z2=1

,A(1,0,0)B(0,1,0)C(0,0,1)A.

I=3AB(z2 y2)dx+(x2 z2)dy+(y2 x2)dz

北大版高等数学习题解答

14.

xy(ydx xdy)L

cos2θcosθ,y=a

π

4

√

4

≤θ≤

π

2a

4

.

(2)

(4)

y2dx+x2dy,L

.

.I=(2y 2x)dxdy,

L

O(0,0),B(1,0),C(0,1)

I=0.

OBC

.

L+(x

2.

+exsiny)dx+(x+excosy)dy,Lr2=cos2θ

π √

xx

DL.I=(1+ecosy ecosy)dxdy=

△OBC

.

D

4

(5)√

L+

(exsiny+sinx 8y)dx+(excosy siny)dy,

L

0≤y≤

8πa

2

.

x=a(1 cost)cost,y=a(1 cost)sint(0≤t≤2π).

2π2

2

L.S=120a(1 cost)dt=

3

(2)

.

y

2

2x

=

1

(x y)

22x

+xy

y

+xy

1

=

( exsiny)

北大版高等数学习题解答

5.

AB

(x4+4xy3)dx+(6x2y2 5y4)dy.

,

A( 2, 1),B(3,0),AB

.(x4+4xy3)dx+(6x2y2 5y4)dy=d(1

5235(3,0)x+2xy y)|( 2, 1)=62.5

7.a,b,

(y2+2xy+ax2)dx (x2+2xy+by2)dy

(x2+y2)2

x

,Q= x

x y

2

+2xy+by2

y

=P,u=x y (y)=C.u=

=

Q

y=

x2+2xy+by2

y

=Q,

√√

√

√

x2+y2+xy).

2.

9.

.(x2+y)dx+(x y2)dy=d(1

I=(

(x2+y)dx+(x y2)dy,ABD

AB

A(0,0)B(1,1)

y3=x2.

=1.

3y

3

),I=(1

3y

3(1,1)

)|(0,0)

10.

.

:

cos(n,x),cos(n,y)L. .L+[Pcos(n,x)+Qcos(n,y)]ds=L+Pdy Qdx=( P

, L,P(x,y), Q(x,y)D

( P y)dσ,L+[Pcos(n,x)+Qcos(n,x)]ds=

D

11..

D

y

)dσ.,nL

L+

L

[xcos n,i +ycos n,j ]ds,L

.

D.I=L+xdy ydx=2dσ=

D

,

D

u(x,y),v(x,y)

.: (1)v△udσ=L+v u

12.

D

,LD

,

v x

y

v

n

u

L

D

(2)

cos n,i ds,dx= cos n,j ds, uL+v

.

D

D

.

(u△v v△u)dσ=v△udσ+

x)+

(

D

D

( u x+

y))dσ,

L+

(u v

n)ds.

u

y)dσ=

=

,

DL+

(v

2

u

=

v u

y2

+

u

x

+

u

y)dσ

=

L+

v u

ydx, y

dy=

cos n,j ds=

北大版高等数学习题解答

13.(1)

u

2

u(x,y)u

D

,

u(x,y)

,

L+

y2 u

=0.

:+( u

2 x)

2 x)

x,

u

+( u

2 x)

+( u