高数第六版上册课后习题答案第六章

习题6-2 1 求图6-21 中各画斜线部分的面积
(1)
解 画斜线部分在x 轴上的投影区间为[0 1] 所求的面积为
6
1]2132[)(102231
0=-=-=⎰x x dx x x A . (2)
解法一 画斜线部分在x 轴上的投影区间为[0 1] 所求的面积为 1|)()(101
0=-=-=⎰x x e ex dx e e A
解法二 画斜线部分在y 轴上的投影区间为[1 e ] 所求的面积为
1)1(|ln ln 1
11=--=-==⎰⎰e e dy y y ydy A e
e e
(3)
解 画斜线部分在x 轴上的投影区间为[-3 1] 所求的面积为
3
32]2)3[(1
32=--=⎰-dx x x A
(4)
解 画斜线部分在x 轴上的投影区间为[-1 3] 所求的面积为
3
32
|)313()32(31323
12=-+=-+=--⎰x x x dx x x A
2. 求由下列各曲线所围成的图形的面积:
(1) 221
x y =与x 2+y 2=8(两部分都要计算);
解:
3
8
8282)218(220220*********--=--=--=⎰⎰⎰⎰dx x dx x dx x dx x x A
3
4238c o s 16402
+=-=⎰ππ
tdt . 34
6)22(122-=-=ππS A .
(2)x
y 1
=与直线y =x 及x =2;
解:
所求的面积为
⎰-=-=2
12ln 2
3)1(dx x x A .
(3) y =e x , y =e -x 与直线x =1; 解:
所求的面积为
⎰-+=-=-1021
)(e
e dx e e A x x .
(4)y =ln x , y 轴与直线y =ln a , y =ln b (b >a >0). 解
所求的面积为
a b e dy e A b
a y b
a y -===⎰ln ln ln ln
3. 求抛物线y =-x 2+4x -3及其在点(0, -3)和(3, 0)处的切线所围成的图形的面积. 解:
y '=-2 x +4.
过点(0, -3)处的切线的斜率为4, 切线方程为y =4(x -3). 过点(3, 0)处的切线的斜率为-2, 切线方程为y =-2x +6.
两切线的交点为)3 ,2
3
(, 所求的面积为
49]34(62[)]34(34[23
023
2
32=-+--+-+-+---=⎰⎰dx x x x x x x A .
4. 求抛物线y 2
=2px 及其在点),2(p p 处的法线所围成的图形的面积.
解
2y ⋅y '=2p
在点),2
(p p
处 1)
,2(==
'p p y p y 法线的斜率k =-1 法线的方程为)2(p
x p y --=- 即y p
x -=
2
3 求得法线与抛物线的两个交点为),2(p p 和)3,29
(p p -
法线与抛物线所围成的图形的面积为
2332323
16)612123()223(p y p y y p dy p y y p A p p
p
p =--=--=--⎰
5. 求由下列各曲线 所围成的图形的面积; (1) =2a cos θ
解:
所求的面积为
⎰⎰==-202222
2c o s 4)c o s 2(21π
ππθθθθd a d a A =πa 2.
(2)x =a cos 3t , y =a sin 3t ; 解
所求的面积为
⎰⎰⎰===204220
2
330s i n c o s 34)c o s ()s i n (44π
πt d t t a t a d t a y d x
A a
220620428
3]s i n s i
n [12a tdt tdt a ππ
π
=-=⎰
⎰
(3) =2a (2+cos θ ) 解
所求的面积为
2202220218)cos cos 44(2)]cos 2(2[2
1a d a d a A πθθθθθπ
π=++=+=⎰⎰
6. 求由摆线x =a (t -sin t ), y =a (1-cos t )的一拱(0 t 2π)与横轴 所围成的图形的面积.
解:
所求的面积为 ⎰⎰⎰
-=--==a
a
a dt t a dt t a t a ydx A 20222020
)cos 1()cos 1()cos 1(π
π22023)2
c o s 1c o s 21(a dt t t
a a
=++-=⎰. 7. 求对数螺线 =ae θ(-π≤θ≤π)及射线θ=π所围成的图形面积 解
所求的面积为
)(421)(21222
222ππππθππθθθ----===⎰⎰e e a d e a d ae A
8. 求下列各曲线所围成图形的公共部分的面积. (1) =3cos θ 及 =1+cos θ
解
曲线 =3cos θ 与 =1+cos θ 交点的极坐标为)3,23(πA , )3,23(π-B . 由对称性, 所求的面积为
πθθθθπ
ππ4
5])c o s 3(21)c o s 1(21[2232302=++=⎰⎰d d A . (2)θρsin 2=及θρ2cos 2=
解
曲线θρsin 2=与θρ2cos 2=的交点M 的极坐标为M )6
,22(π 所求的面积为
2
316]2cos 21)sin 2(21[246602-+=+=⎰⎰πθθθθπππd d A
9. 求位于曲线y =e x 下方, 该曲线过原点的切线的左方以及x 轴上方之间的图形的面积.
解 设直线y =kx 与曲线y =e x 相切于A (x 0 y 0)点 则有
⎪⎩⎪
⎨⎧=='==k
e x y e y kx y x x 00)(0000
求得x 0=1 y 0=e k =e 所求面积为
2
1ln 21)ln 1(00020e dy y y y y y e dy y y e e e e
e
=⋅+-=-⎰⎰
10. 求由抛物线y 2=4ax 与过焦点的弦所围成的图形的面积的最小值. 解 设弦的倾角为α. 由图可以看出, 抛物线与过焦点的弦所围成的图形的面积为 10A A A +=.
显然当2πα=时, A 1=0; 当2πα<时, A 1>0.
因此, 抛物线与过焦点的弦所围成的图形的面积的最小值为 203
03
83822a x a dx ax A a a
===⎰.
11. 把抛物线y 2=4ax 及直线x =x 0(x 0>0)所围成的图形绕x 轴旋转, 计算所得旋转体的体积.
解 所得旋转体的体积为
2
0020
222400
x a x a axdx dx y V x
x x ππππ====⎰⎰
12. 由y =x 3 x =2 y =0所围成的图形 分别绕x
轴及y 轴旋转 计算所得两个旋转体的体积 解 绕x 轴旋转所得旋转体的体积为 ππππ7
128712
072
062
02
====⎰⎰x dx x dx y V x
绕y 轴旋转所得旋转体的体积为 ⎰⎰-=-⋅⋅=8
03
2
8
02
2
3282dy y dy x V y ππππ
πππ5
6453328035=-=y
13. 把星形线3/23/23/2a y x =+所围成的图形 绕x 轴旋转 计算所得旋转体的体积
解 由对称性 所求旋转体的体积为 dx x a dx y V a
a
⎰⎰-==03
323
202)(22ππ
3023
43
23
23
4
2
105
32)33(2a dx x x a x a a a
ππ=-+-=⎰
14. 用积分方法证明图中球缺的体积为)3(2H R H V -=π
证明 ⎰
⎰
---==R
H
R R H
R dy y R dy y x V )()(222
ππ
)3()31(232H R H y y R R
H R -=-=-ππ
15. 求下列已知曲线所围成的图形, 按指定的轴旋转所产生的旋转体的体积:
(1)2x y =, 2y x =, 绕y 轴;
解 ππππ10
3)5121()(1
0521
0221
0=-=-=⎰⎰y y dy y ydy V .
(2)a x a y ch = x =0 x =a y =0 绕x 轴 解 ⎰⎰⎰===102302
2
02
ch ch )(udu a au x dx a
x a dx x y V a
a
πππ令
10
22310223)2
1221(4)2(4u u u
u e u e a du e e a ---+=++=⎰ππ )2sh 2(4
3
+=
a π (3)16)5(22=-+y x , 绕x 轴.
解 ⎰⎰------+=4
4
224
4
2
2)165()165(dx x dx x V ππ
24
21601640π⎰=-=dx x .
(4)摆线x =a (t -sin t ), y =a (1-cos t )的一拱, y =0, 绕直线y =2a .
解 ⎰⎰--=ππππa a dx y a dx a V 202202)2()2( ⎰----=πππ20223)sin ()]cos 1(2[8t t da t a a a
232023237sin )cos 1(8ππππa tdt t a a =+-=⎰. 16 求圆盘222a y x ≤+绕x =-b (b >a >0)旋转所成旋转体的体积.
解 ⎰⎰------+=a
a
a
a dy y a
b dy y a b V 2222
22
)()(ππ
22
2228ππb a dy y a b a
=-=⎰
.
17 设有一截锥体 其高为h 上、下底均为椭圆 椭圆的轴长分别为2a 、2b 和2A 、2B 求这截锥体的体积
解 建立坐标系如图 过y 轴上y 点作垂直
于y 轴的平面 则平面与截锥体的截面为椭圆 易得其长短半轴分别为
y h a A A -- y h b B B --
截面的面积为π)()(y h b B B y h a A A --⋅--
于是截锥体的体积为
])(2[6
1)()(0bA aB AB ab h dy y h b B B y h a A A V h
+++=--⋅--=⎰ππ
18 计算底面是半径为R 的圆, 而垂直于底面上一条固定直径的所有截面都是等边三角形的立体体积.
解 设过点x 且垂直于x 轴的截面面积为A (x ),
由已知条件知, 它是边长为x R -2的等边三角形的面积, 其值为
)(3)(22x R x A -=, 所以 3223
34)(3R dx x R V R
R
=-=⎰
-.
19. 证明 由平面图形0≤a ≤x ≤b 0≤y ≤f (x )绕y 轴旋转所成的旋转体的体积为
⎰=b
a dx x xf V )(2π
证明 如图 在x 处取一宽为dx 的小曲边梯形 小曲边梯形绕y 轴旋转所得的旋转体的体积近似为2 x ⋅f (x )dx
这就是体积元素 即 dV =2 x ⋅f (x )dx
于是平面图形绕y 轴旋转所成的旋转体的体积为 ⎰⎰==b
a
b
a
dx x xf dx x xf V )(2)(2ππ
20. 利用题19和结论 计算曲线y =sin x (0≤x ≤ )和x 轴所围成的图形绕y 轴旋转所得旋转体的体积
解 200
2)sin cos (2cos 2sin 2πππππ
π
π=+-=-==⎰⎰x x x x xd xdx x V
21. 计算曲线y =ln x 上相应于83≤≤x 的一段弧的长度. 解 ⎰⎰
⎰
+=+='+=83283
2
83
2
1)1(1)(1dx x
x dx x dx x y s ,
令t x =+21, 即12-=t x , 则 23ln 2111111
1322323
222232
2+=-+=-=-⋅-=⎰⎰⎰⎰
dt t dt dt t t dt t t
t t s .
22. 计算曲线)3(3x x y -=上相应于
1≤x ≤3的一段弧的长度 解 x x x y 3
1-= x x y 2121-='
x x y 4121412+-=' )1(2112x
x y +='+
所求弧长为
3432)232(21)1(213
131-=+=+=⎰x x x dx x
x s
23. 计算半立方抛物线32)1(32-=x y 被抛物线32x y =截得的一段弧
的长度
解 由⎪⎩
⎪⎨⎧=-=3)1(3223
2x y x y 得两曲线的交点的坐标为)36 ,2( )
3
6 ,2(-
所求弧长为⎰'+=2
1212dx y s
因为 2
)1(22-='x y y y
x y 2
)1(-=' )1(23)1(32)1()1(34
2
42
-=--=-=
'x x x y x y
所以 ]1)25[(98)13(13232)1(23122321
2
1
-=--=-+=⎰⎰
x d x dx x s
24. 计算抛物线y 2=2px 从顶点到这曲线上的一点M (x , y )的弧长.
解 ⎰⎰
⎰
+=+='+=y y
y
dy y p p dy p y dy y x s 0
220
20
21)(1)(1
y y p y p y p y p 022222])ln(22[1++++=
p
y p y p
y p p y 2
222ln
22++++=. 25. 计算星形线t a x 3cos =, t a y 3sin =的全长.
解 用参数方程的弧长公式. dt t y t x s ⎰'+'=20
22)()(4π
⎰⋅+-⋅=20
2222]cos sin 3[)]sin (cos 3[4π
dt t t a t t a
a tdt t 6cos sin 1220
==⎰π
.
26. 将绕在圆(半径为a )上的细线放开拉直 使细线与圆周始终相切 细线端点画出的轨迹叫做圆的渐伸线 它的方程为 )sin (cos t t t a x +=
)cos (sin t t t a y -=
计算这曲线上相应于t 从0变到 的一段弧的长度
解 由参数方程弧长公式 ⎰
⎰+='+'=π
π0
220
2
2
)sin ()cos ()]([)]([dt t at t at dt t y t x s
202
ππ
a
tdt a ==⎰
27. 在摆线x =a (t -sin t ) y =a (1-cos t )上求分摆线第一拱成1
3的点的坐标
解 设t 从0变化到t 0时摆线第一拱上对应的弧长为s (t 0) 则 ⎰
⎰
+-='+'=0
00
220
220]sin [)]cos 1([)]([)]([)(t t dt t a t a dt t y t x t s
)2
cos 1(42sin 2000t
a dt t a t -==⎰
当t 0=2 时 得第一拱弧长s (2 )=8a 为求分摆线第一拱为
1 3的点为A (x y ) 令 a t
a 2)2
cos 1(40=-
解得3
20π=t 因而分点的坐标为
横坐标a a x )2332()32sin 32(-=-=πππ
纵坐标a a y 2
3)32cos 1(=-=π
故所求分点的坐标为)2
3 ,)2332((a a -π
28. 求对数螺线θρa e =相应于自θ=0到θ=ϕ的一段弧长. 解 用极坐标的弧长公式. θθθρθρϕ
θθϕ
d a
e e d s a a ⎰⎰+='+=0
220
2
2
)()()()(
)1(112
2-+=+=⎰
θϕ
θ
θa a e a
a d e a . 29. 求曲线 =1相应于自43=θ至3
4
=θ的一段弧长
解 按极坐标公式可得所求的弧长 ⎰
⎰
-+='+=344
322234
4
32
2
)1()1()()(θθ
θθθρθρd d s
23ln 12511344
322+=+=⎰θθθd
30. 求心形线 =a (1+cos θ )的全长. 解 用极坐标的弧长公式. θθθθθρθρπ
π
d a a d s ⎰⎰
-++='+=0
2220
2
2
)sin ()cos 1(2)()(2
a d a 82
cos 40==⎰π
θθ.
习题6-3
1. 由实验知道, 弹簧在拉伸过程中, 需要的力F (单位: N )与伸长量s (单位: cm)成正比, 即F =ks (k 为比例常数). 如果把弹簧由原长拉伸6cm , 计算所作的功.
解 将弹簧一端固定于A , 另一端在自由长度时的点O 为坐标原点, 建立坐标系. 功元素为dW =ksds , 所求功为 182
16
026
0===⎰s k ksds W k(牛⋅厘米). 2. 直径为20cm 、高80cm 的圆柱体内充满压强为10N/cm 2的蒸汽. 设温度保持不变, 要使蒸汽体积缩小一半, 问需要作多少功？ 解 由玻-马定律知:
ππ80000)8010(102=⋅⋅==k PV .
设蒸气在圆柱体内变化时底面积不变, 高度减小x 厘米时压强 为P (x )牛/厘米2, 则
ππ80000)]80)(10[()(2=-⋅x x P , π
-=80800
)(x P .
功元素为dx x P dW )()10(2⋅=π,
所求功为 2ln 8008018000080800)10(40040
2
ππππ
π=-=-⋅⋅=⎰⎰
dx dx W (J). 3. (1)证明: 把质量为m 的物体从地球表面升高到h 处所作的功是
h
R mgRh
W +=
, 其中g 是地面上的重力加速度, R 是地球的半径;
(2)一颗人造地球卫星的质量为173kg , 在高于地面630km 处进入轨道. 问把这颗卫星从地面送到630的高空处, 克服地球引力要作多少功？已知g =9.8m/s 2, 地球半径R =6370km .
证明 (1)取地球中心为坐标原点, 把质量为m 的物体升高的功元素为
dy y kMm dW 2
=, 所求的功为 )
(2h R R mMh k dy y kMm W h R R
+⋅==⎰
+. (2)5
3
33
2411
1075.910)6306370(106370106301098.51731067.6⨯=⨯+⨯⨯⨯⨯⨯⋅⨯=-W (kJ).
4. 一物体按规律3ct x =作直线运动, 媒质的阻力与速度的平方成正比. 计算物体由x =0移至x =a 时, 克服媒质阻力所作的功. 解 因为3ct x =, 所以
23)(cx t x v ='=, 阻力4
229t kc kv f -=-=. 而32)(c
x t =, 所以
3432342
9)(9)(x kc c
x kc x f -=-=. 功元素dW =-f (x )dx , 所求之功为 37
320
3
4320
3
43
20
7
2799)]([a kc dx x kc
dx x kc dx x f W a a
a
===-=⎰
⎰⎰. 5. 用铁锤将一铁钉击入木板, 设木板对铁钉的阻力与铁钉击入木板的深度成正比, 在击第一次时, 将铁钉击入木板1cm . 如果铁锤每次打击铁钉所做的功相等, 问锤击第二次时, 铁钉又击入多少？ 解 设锤击第二次时铁钉又击入h cm , 因木板对铁钉的阻力f 与铁钉击入木板的深度x (cm)成正比, 即f =kx , 功元素dW =f dx =kxdx , 击第一次作功为
k kxdx W 2
11
01==⎰,
击第二次作功为
)2(2
12112h h k kxdx W h
+==⎰+. 因为21W W =, 所以有 )2(21212h h k k +=, 解得12-=h (cm).
6. 设一锥形贮水池, 深15m , 口径20m , 盛满水, 今以唧筒将水吸尽, 问要作多少功？
解 在水深x 处, 水平截面半径为x r 3
2
10-=, 功元素为
dx x x dx r x dW 22)3210(-=⋅=ππ,
所求功为
⎰-=15
02)3
210(dx x x W π
⎰+-=15
032)9
440100(dx x x x π =1875(吨米)=57785.7(kJ).
7. 有一闸门, 它的形状和尺寸如图, 水面超过门顶2m . 求闸门上所受的水压力.
解 建立x 轴, 方向向下, 原点在水面. 水压力元素为
xdx dx x dP 221=⋅⋅=, 闸门上所受的水压力为
2125
225
2===⎰x xdx P (吨)=205. 8(kN).
8. 洒水车上的水箱是一个横放的椭圆柱体, 尺寸如图所示. 当水箱装满水时, 计算水箱的一个端面所受的压力.
解 建立坐标系如图, 则椭圆的方程为 11)4
3()43(22
22=+-y x . 压力元素为
dx x x dx x y x dP 22)43()43(38)(21--⋅=⋅⋅=,
所求压力为 ⎰⎰
-⋅⋅+=--⋅=222
3
22cos 4
3cos 43)sin 1(4338)43()43(38π
πtdx t t dx x x P ππ
169cos 49202==⎰tdx (吨)=17.3(kN). (提示: 积分中所作的变换为t x sin 4
343=-)
9. 有一等腰梯形闸门, 它的两条底边各长10m 和6m , 高为20m . 较长的底边与水面相齐. 计算闸门的一侧所受的水压力. 解 建立坐标系如图. 直线AB 的方程为
x y 1015-=,
压力元素为
dx x x dx x y x dP )5110()(21-⋅=⋅⋅=,
所求压力为
1467)5
110(20
0=-⋅=⎰dx x x P (吨)=14388(千牛).
10. 一底为8cm 、高为6cm 的等腰三角形片, 铅直地沉没在水中, 顶在上, 底在下且与水面平行, 而顶离水面3cm , 试求它每面所受的压力.
解 建立坐标系如图.
腰AC 的方程为x y 3
2
=, 压力元素为
dx x x dx x x dP )3(34322)3(+=⋅⋅⋅+=,
所求压力为
168)2
331(34)3(346
0236
0=+=+=⎰x x dx x x P (克)=1.65(牛).
11. 设有一长度为l 、线密度为μ的均匀细直棒, 在与棒的一端垂直距离为a 单位处有一质量为m 的质点M , 试求这细棒对质点M 的引力.
解 建立坐标系如图. 在细直棒上取一小段dy , 引力元素为 dy
y
a Gm y a dy m G dF 222
2+=+⋅
=μ
μ, dF 在x 轴方向和y 轴方向上的分力分别为 dF r a dF x -=, dF r
y
dF y =.
2202222022)(1)(l
a a l Gm dy y a y a aGm dy y a Gm r a F l l
x +-=++-=+⋅-=⎰⎰μμμ, )11()(12
2
02222022l a a Gm dy y a y a Gm dy y a Gm r y F l l
y +-=++=+⋅=⎰⎰μμμ. 12. 设有一半径为R 、中心角为 ϕ 的圆弧形细棒, 其线密度为常数 μ . 在圆心处有一质量为m 的质点F . 试求这细棒对质点M 的引力.
解 根据对称性, F y =0. θμcos 2
⋅⋅⋅=
R
ds
m G dF x
θθμθθμd R
Gm R Rd Gm cos cos )(2=⋅=, θθμϕ
ϕd R Gm F x ⎰-=22
cos 2
sin 2cos 220ϕμθθμϕR Gm d R Gm ==⎰. 引力的大小为
2
sin 2ϕμR Gm , 方向自M 点起指向圆弧中点.
总 习 题 六
1. 一金属棒长3m , 离棒左端xm 处的线密度为11)(+=x x ρ (kg/m ). 问x 为何值时, [0, x ]一段的质量为全棒质量的一半? 解 x 应满足⎰⎰
+=+30011211
1dt t dt t x . 因为212]12[1
100-+=+=+⎰x t dt t x x , 1]12[2111213030=+=+⎰t dt t , 所以 1212=-+x ,
4
5=x (m). 2. 求由曲线ρ=a sin θ, ρ=a (cos θ+sin θ)(a >0)所围图形公共部分的面积.
解
⎰++⋅=432222)sin (cos 21)2(21π
πθθθπd a a S
2
4322241)2sin 1(28a d a a -=++=⎰πθθπππ
3. 设抛物线c bx ax y ++=2通过点(0, 0), 且当x ∈[0, 1]时, y ≥0. 试确定a 、b 、c 的值, 使得抛物线c bx ax y ++=2与直线x =1, y =0所围图形的面积为9
4, 且使该图形绕x 轴旋转而成的旋转体的体积最小. 解 因为抛物线c bx ax y ++=2通过点(0 0) 所以c =0 从而 bx ax y +=2
抛物线bx ax y +=2与直线x =1, y =0所围图形的面积为
23)(102b
a dx bx ax S +=+=⎰ 令94
23=+b a 得968a
b -=
该图形绕x 轴旋转而成的旋转体的体积为
)235()(22102
2ab b a dx bx ax V ++=+=⎰ππ )]968(2)968(315[22a a a a -+-+=π
令0)]128(181********[=-+-⋅+2=a a a d dV π 得3
5-=a 于是b =2
4. 求由曲线23x y =与直线x =4, x 轴所围图形绕y 轴旋转而成的旋转体的体积.
解 所求旋转体的体积为
πππ7512722240274023=⋅=⋅=⎰x dx x x V
5. 求圆盘1)2(22≤+-y x 绕y 轴旋转而成的旋转体的体积. 解 )2(1223
12⎰--⋅⋅=dx x x V π
22224cos )sin 2(4 sin 2πππ
π=+=-⎰-tdt t t x 令 6. 抛物线22
1x y =被圆322=+y x 所需截下的有限部分的弧长.
解 由⎪⎩⎪⎨⎧==+222213x y y x 解得抛物线与圆的两个交点为)1 ,2(- )
1 ,2( 于是所求的弧长为
202220
2])1ln(2112[212x x x x dx x s ++++=+=⎰ )32ln(6++=
7. 半径为r 的球沉入水中, 球的上部
与水面相切, 球的比重与水相同, 现将球
从水中取出, 需作多少功?
解 建立坐标系如图 将球从水中取
出时 球的各点上升的高度均为2r 在x 处取一厚度为dx 的薄片 在将球从水中取出的过程中 薄片在水下上升的高度为r +x 在水上上升的高度为r -x 在水下对薄片所做的功为零 在水上对薄片所做的功为
dx x r x r g dW ))((22--=π
对球所做的功为
g r x d x r x r g W r r 22234))((ππ=--=⎰-
8. 边长为a 和b 的矩形薄板, 与液面成α 角斜沉于液体内, 长边平行于液面而位于深h 处, 设a >b , 液体的比重为ρ, 试求薄板每面所受的压力.
解 在水面上建立x 轴 使长边与x 轴在同一垂面上 长边的上端点与原点对应 长边在x 轴上的投影区间为[0 b cos ] 在x 处x 轴到薄板的距离为h +x tan 压力元素为
dx x h ga dx a x h g dP )tan (cos cos )tan (αα
ρααρ+=⋅⋅+⋅= 薄板各面所受到的压力为
)sin 2(2
1)tan (cos cos 0αρααραb h gab dx x h ga P b +=+=⎰ 9. 设星形线t
a x 3cos = t a y 3sin =上每一点处的线密度的大小等于该点到原点距离的立方, 在原点O 处有一单位质点, 求星形线在第一象限的弧段对这质点的引力.
解 取弧微分ds 为质点 则其质量为
ds y x ds y x 322322)()(+=+ 其中tdt t a dt t a t a ds cos sin 3])sin [(])cos [(2323='+'=
设所求的引力在x 轴、y 轴上的投影分别为F x 、F y 则有
⎰+⋅++⋅⋅=202222322)()(1πds y x x y x y x G F x 2204253sin cos 3Ga tdt t Ga ==⎰π
⎰+⋅++⋅⋅=2
02
222322)()(1πds y x y y x y x G F x 2204253sin cos 3Ga tdt t Ga ==⎰π 所以)5
3 ,53(22Ga Ga =F
（英文版）
easily blame, to prevent the broken window effect. Supervise the leading cadres to play an exemplary role, take the lead in the strict implementation of the < code > and < rule >, lead to safeguard the solemnity and authority of the party discipline, ensure that the party discipline and the laws and regulations for implementation in place. Throughout the discipline in the daily supervision and management , strengthen supervision and inspection, from the thorough investigation of violations of discipline behavior. Strengthen to key areas, key departments and key projects as well as the masses reflect the concentration of the units and departments for supervision. - strengthening supervision, discipline inspection and supervision of cadres to set an example for compliance with the < code > and < rule > is a man must be hexyl, blacksmith needs its own hardware. Discipline inspection organs as the executor of the party discipline, and supervisor of the defenders, for its supervision must be more strictly, discipline inspection and supervision of cadres to firmly establish the awareness of Party Constitution, sense of discipline and rules consciousness, politics loyalty, sense obey. Action speak Ji Ordinance to set an example of the regulations of the rule of law, strengthen supervision and accept the supervision of the firmness and consciousness, do comply with < > and < >. To firmly establish the discipline must first be disciplined, the supervisor will be subject to the
supervision of "concept, and consciously safeguard and implement party compasses party, take the lead in practicing" three strict real strict, so loyal, clean, play. To be good at learning, the Constitution and the < code > as morality, politics and brought to fruition; to implement < >, do not want to, dare not, not with disciplinary ruler to supervision; to discipline a ruler, often the control inspection, and consciously in the ideological red line to draw the row Ming Good accumulation is indeed the bottom line, so that the heart has fear, said to have quit, the line has ended. Attached: indifferent to heart, calmly to the table in our life, there are many unpredictable things will happen, some good, some bad things, we cannot control is powerless to stop, but with time, you will find in life sometimes turns out to be not good, some bad things finally turned out to be a good thing, but then we muddy however did not know, this is the life teach us things. 1, life can be complex, can also be simple. Want simple life of precipitation, to have enough time to reflect, to make Become more perfect. Life is the most important thing is not to win, but the struggle; not to have conquered, but to have fought well. 2, the plain is the background of life. Live a plain life, give up on themselves is not a coward, but the wise answers; not disillusioned after the heart, such as ashes, but experience the storm after the enlightenment; not unrewarding perfunctorily, but calm attitude of life of unrestrained self-confidence. Plain living, there is no noise noisy, no earthly troubles, more did not fill in the discontent of desire,
some just a calm, a calm. 3, memory of heart will not good things to erase the, life is a When no movie, pain is a beginning, the struggle is a kind of process, death is a kind of ending. Give up this giving up is the helpless, do not give up the abandoned, do not give up this giving up is ignorance, do not give up should not give up is persistent. 4, a thing figured is heaven, think impassability is hell. Since the living, to live better. Sometimes we because of too narrow-minded, too care around the chores and penny wise and pound foolish, not worth the candle. Some things to attract trouble and worry, completely depends on how we look at and deal with it. Don't always take everything back to things, and don't get into a blind alley, don't want to face, don't be narrow-minded. Poke to care, is a kind of open-minded, a free and easy. 5, I am not afraid of others behind me a knife, I afraid to look back and see stab me, is my intention to treat people; I am not afraid of the truth to tell the best friend, I'm afraid he turned to it as a joke to tell don't 6, when we are in a positive frame of mind, you will find many good things; and when we are in a negative state of mind, you will find many depressed things; life happy and worry, all is you of life attitude, optimistic, good luck; loss of sink, Eritrea company. When you are in adversity, may wish to change a point of view to think everything over to the good Think, because good mentality decided the fate of the! 7, people are tired, rest; heart tired, calm. Grow up, mature, this society read. Tired and sad, squat down, to their a hug. Because the world no one can
sympathize with you, have mercy on you. Y ou cry, tears is your own; you pain, no one can understand. Then you only tears to smile. 8, each people have youth,
Each youth are a story, the life of the world never gets easier, I want what, wish the world all know, as has been the same; now want anything, for fear that others know, or like to lose the same. 9, the heart move, everything in the world is followed by birth, Rangrang, important thing is often the most difficult to open one's mouth, because words will reduce its importance; to let strangers people care about your life in the good things, the original is not easy 10, do not blame, do not laugh at who, also don't envy who. Like a person is a kind of feeling, not like a person is true. The truth is easy to explain, I feel Is unspeakable. The best travel life is that you in a strange place found a long lost touched. 11, happy life not in the bustling in, and in the peace of mind; no matter how many grievances, how uncomfortable, and ultimately to heal themselves or their own, others may got you to comfort, but never know your heart is how wanjianchuanxin. 12, ma'am, like a movie, learn to appreciate, learn to be grateful, learn tolerance, and goodness, helping others. Instead of accusing the society, as into one; and an exception is better to give than to what 13, don't envy him A sum of, don't lose your life and the life, respectively is: the former is a we experienced cannot escape in a day finally will last minute, while the latter is our persistent, we want to cherish the memory of those people and things.
14, learn to smile, learn to strong, the world you know so many people, so many people and you are, you cannot change also can't let everyone like you, so also do not want to do. Life is too short to go crazy to love to go to waste, to chase the dream to regret. 15, when temper, a blessing to go. A wounding elegant people, the key is to control their own emotions. With the mouth is the most stupid behavior. A control negative emotions than a can take a city more powerful water flow slow, language is expensive. People spent two years of time to learn to speak, but to spend a few years time to shut up. That is a kind of ability, that is a kind of wisdom. 16, life is not perfect, sometimes, growth is not a cry, not an eyeful of tears, there is no trace of emotion, there is no gleam of hope, no desire, no action, no static, there is only one kind of downward sinking feeling, sink A murky? 6? 7? 6? 7 sink? 6? 7? 6? 7 toward the bottom of the sink. 17, in some way, do not go, you will not know the other side scenery is beautiful. To you is not good, you do not mind too much, no one has an obligation to you; you learn knowledge, is you have weapons, you can start from scratch, but not unarmed; how do you treat people, does not represent how others treat you, if cannot see through this point, only inviting worry. 18, time is like a sponge in the water, as long as you are willing to squeeze, the total water is still there. Every life, after the ups and downs The best test of live, to life, survival and continuation, do not stop the struggle in the joys and sorrows of life on the road, so that different soul to bear life beat, acceptance of
suffering. 19, indifferent to heart, calmly in table, elegant and comfortable life, do not take what is so important. The pursuit will be disappointed; to be alive, you will have trouble. Life is the most afraid of what all want to care about, but also what all grasp is not firm, without scenery, separated populations, such as not to desire, all docked in the fate of the end. Why is too rigid, the natural, to go stay not to live, let go of obsession, revel is 20, if the fate of the broken Hopes of sailing, please don't despair, the coast is still, if the fate of the withered petals of the beautiful, please do not sink, the spring is still, life will always be endless trouble, please don't helpless, because they are still alive, is still a dream, the sun still, we still. Lost, keep memories; to get, must work to; but the most important is good to cherish their own. 21, life, select the complex, is to choose the pain; choose a simple, is choose to be happy. The complex world like a
Significance of pride. Hope is the ornate palace, outside people admiring the magnificent, living in the deep knowledge of living it to pay the price. Simple world as a simple log cabin outside ridiculed shabby, the heart is willing to go live to know the joy. Suffering and joy is their own choice. 22, learn how to use a single powerful heart, let the past be the past, let the future come. Life is really the end of the end of an eagle is flying wings, life is constantly pursuit. Don't miss to regret, don't wait for old just miss. Time to return, seize every moment, again painstakingly again tired also Those struggling to fly. 23, life could not Yimapingchuan, even flat
pavement, inevitably there will be a few pieces of roadblocks. Some of the rocks around the past, while others have to move it out. Just move others put the stone is very easy, because the stone from the appearance we can discern; difficult to myself to move away the heart of stone head. Leave time to spend with her, often reflect my heart, so as to remove your heart of stone. 24, everything does not have to be demanding, come to, everything does not have to care about, over the past; failing to do not frown, laugh it laugh. Results Don't demand, do to; life is a simple, calm and peaceful. Always not to choose their own path and regret, life is like a train, the scenery and then the United States will retreat, the passage of time and encounter will eventually drifting further and further away, before is always himself. 25, everyone has a weakness, weakness is true humanity. That has no weakness, a shallow person. That people think there is no weakness, mostly false. Life has shortcomings, there are shortcomings is the real life. That no one regret, or childish or numbness or Self deception. It is in tolerance of weakness and so on to accept, people live happily.
Hello, everyone! I am a party member. The title of my speech is: < study and implement the party's two laws, doing practical play highway. 2015 October 18, the Central Committee of the Communist Party of China promulgated the implementation of the < the probity of the Communist Party of China self-discipline criterion > and < Chinese Communist Party
discipline and punishment regulations. We Heyuan male passers-by to respond positively to the call of the Central Committee of the party, earnestly organize the study "party two regulations", truly grasp the essence and gist, and in their respective positions, to hold the bottom line of the discipline, build a strong ideological line of defense, with the courage to play, the courage to fight tough and fearless spirit, at the crucial moment well to complete the task, with practical action to test the study and implement effect Because of discipline in the * * * * * * * * * * * story. Here, let me to cast a brick to attract jade, speak about our highway. Highway line section of the road surface transformation project, last year "towards the country seized" will be seized one of the items. To complete this arduous task, as a project management office director Comrade, keep in mind from the Communist Party membership, recognize and identify the "bottom line", strict management, and strict adherence to the quality of the project. He not only set an example, honesty and self-discipline, but also requires the management of all the members of the O.K., do not eat the construction unit one meal, do not accept the construction unit a ceremony. In this way, they didn't really dare to adhere to the principle. No comrade, constantly put on reworking an emergency meeting to Comrade Zhen to speak louder, management tube too strict. Remember in Dongguan Street, 400 meters long cement concrete surface layer, because of various reasons, the smoothness of the poor in the bottom cavity, covering film traces and
car imprinting quality problems, * * * inquiries, immediately rushed to the scene to understand and verify the situation, the convening of the management office, the construction units, supervision units, construction units, construction units construction time is tight, the economic loss and other reasons to intercede ******** unmoved. He said, "now a popular word, to the discipline and rules quite in front, there are no rules Radius, you construction team not accordance with the technical specifications, quality problems, it must be to carry out rectification. Engineering quality responsibility be weightier than Mount Tai, if we manage to this matter Pavement quality quantity are placed the matter, we this time to learn two regulations have what use? Still what is the Communist Party? "Finally, in his insisted, the road after rework, to solve these problems. In the construction of the new comrades and the project all the colleagues efforts, after four months of fighting, the project the main project was finally completed and passed inspection. Thousands of miles of ice, thousands of miles Piao, this is a splendid and romantic scene. But snow for the highway, it is a disaster, a serious threat to the traffic safety. This year, a month, a century of cold wave swept from North to south, and the snow blowing to Guangdong, but also to bear the blow To * * * * the highway. In January on the evening of 23, Lianping county city temperatures dropped to minus 2 degrees, a wide range of sudden rain sleet, before and after the provincial S341 line in Jiulianshan Mountain tunnel sections of the road appeared in。