作者LeoRen (Eugene Hecht)
看板trans_math
标题Re: [积分] 请教一题求体积
时间Thu Jun 19 23:58:08 2008
※ 引述《yummy100 (LING)》之铭言：
: 拜托~帮忙解答~
: 求 ∫∫∫(x^2 + y^2 + z^2) 所围体积? R:{ x+y+z=a , x=0, y=0, z=0}
: R
: Ans: a^5 /20
a a-z a-z-y
∫ ∫ ∫ (x^2 + y^2 + z^2) dxdydz
0 0 0
a a-z
=∫ ∫ (1/3)*(a-z-y)^3 + (y^2 +z^2)(a-z-y) dydz
0 0
a
=∫ (1/2)*(z^2)*(a-z)^2 + (1/6)*(a-z)^4 dz
0
= (1/2)*[(a^5)/3 - (a^5)/2 + (a^5)/5] - (-1/30)a^5
= (1/60 + 1/30)a^5
= (1/20)a^5
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◆ From: 118.160.177.227
1F：推 yummy100:请问第二个积分怎麽积的?140.115.234.121 06/20 00:52
2F：→ LeoRen:用看的 (-1/12)(a-z-y)^4 + z^2(a-z)y -118.160.177.227 06/20 01:13
3F：→ LeoRen:(1/2)z^2y^2 + (1/3)(a-z)y^3 - y^4/4118.160.177.227 06/20 01:14
4F：→ LeoRen:看y积分,不要真的去展开,之後同幂项会简化118.160.177.227 06/20 01:18