Interesting MathQ6(SUPER HARD)

設正立方體邊長為2 (方便計算), 球半徑為2
以正立方體中心為(0,0,0), 則8頂點為(1, 1, 1),(1, 1, -1),(1, -1,1), (1, -1, -1)...
8個球的交集為A={ (x,y,z) | (x-a)²+(y-b)²+(z-c)² < 4, 其中a, b, c為 1 or -1}
考慮A的對稱性:
(1)對稱於xy, yz, zx平面 (求 x>0, y>0, z>0部分,即1st octant,再8倍即可)
(2)對稱於x=y(求1st octant,且x>y部分)
設B={ (x,y,z) | (x+1)²+(y+1)²+(z+1)² <4, 且 x>y>0, z>0}, (求B之體積再16倍即可)
={(x,y,z)| x²+y²+z²<4, x>y>0,z>0} -{(x,y,z)| x²+y²+z²<4, y<x<√3 y, 0<y<1, z>0}
volume(B)=(4π/3)*4-∫_[0~1]∫_[y~√3 y] √(4-x²-y²) dx dy
=(16π/3)-(1/3)∫_[π/6~π/4] [8-√(4-csc²t)³] dt
=(16π/3)-(2π/9)+(1/3)∫_[π/6~π/4]√(4-csc²t)³ dt
=(46π/9)+(1/3)∫_[π/6~π/4] √(4-csc²t)³ dt (橢圓積分)
~(46π/9)+(1/3)(0.3767)
Ans: 16*Volume(B)/8= 2*Volume(B)

(因原題球半徑為1,正方體邊長為1, 故除以8)

2011-11-11 16:28:07 補充：
更正:volume(B)=(4π/3)/2-∫_[0~1]∫_[y~√3 y] √(4-x²-y²) dx dy
=(2π/3)-(1/3)∫_[π/6~π/4] [8-√(4-csc²t)³] dt
=(2π/3)-(2π/9)+(1/3)∫_[π/6~π/4]√(4-csc²t)³ dt
=(4π/9)+(1/3)∫_[π/6~π/4] √(4-csc²t)³ dt
~(4π/9)+(1/3)(0.3767)

2011-11-11 16:33:13 補充：
還有錯,晚上再Check!

2011-11-11 16:56:00 補充：
B={(x,y,z)| x²+y²+z²<4, y<√3 y, 0<1, 1





2011-11-11 17:00:02 補充：
B={ (x,y,z) | x²+y²+z² < 4, 且 x > y > 0, z > 1}, (求B之體積再16倍即可)
volume(B)=∫_[0~1]∫_[y~√3 y] √(4-x²-y²) -1 dx dy
=(1/3)∫_[π/6~π/4] [8-√(4-csc²t)³] dt - (√3 -1)/2
=(2π/9)-(1/3)∫_[π/6~π/4]√(4-csc²t)³ dt - (√3 -1)/2
Ans: (4π/9)-(2/3)∫_[π/6~π/4]√(4-csc²t)³ dt - √3 +1 ~ 0.41308

2011-11-11 17:33:09 補充：
最後更正:
B={ (x,y,z) | x²+y²+z² < 4, 且 x > y > 1, z > 1}, (求B之體積再16倍即可)
vol(B)=∫_[a~π/4]∫_[csct~√3] [√(4-r²) -1] rdr dt ( tana= 1/√2 )
Ans: 2vol(B)~ 0.0152

2011-11-11 17:48:18 補充：
B={ (x,y,z) | (x+1)²+(y+1)²+(z+1)² < 4, 且 x > y > 0, z > 0},
={ (x,y,z) | x²+y²+z² < 4, 且 x > y > 1, z > 1}

What is your teacher's final answer?

2011-11-11 16:17:02 補充：
Around 1/9?

2011-11-11 17:18:38 補充：
Yes some mistake!!something like 1/66

You can TRY TRY!

It's still very difficult even you use integration!

2011-11-11 16:08:24 補充：
Actually, he didn't give me the ans, but he agree with my method...

And I've check few times, and the ans the very make sense.

2011-11-11 16:16:27 補充：
林春生，

在http://hk.knowledge.yahoo.com/question/question?qid=7011110200186

你如何算出126.04?

2011-11-11 16:19:10 補充：
Impossible!

The answer is much smaller!

Do you really understand this Q?

2011-11-11 17:19:37 補充：
Umm...GD JOB...

2011-11-11 17:20:50 補充：
I asked sth to you in your mailbox

2011-11-11 17:49:38 補充：
ok...........

我想到一個愚蠢的方法,set個coordinate plane 跟住用 integration...