[微積] 重積分
x y z
求 (──)^1/2 + (──)^1/2 + (──)^1/2 < 1 之體積
a b c =
我的作法是
令 x = au^4 , y = bv^4 , z = cw^4
|J| = 64abc(uvw)^3
再令 u = rsin(f)cos(g)
v = rsin(f)sin(g)
w = rcos(f)
(抱歉,因為太長,我把式子分兩行)
2兀 兀 1
V = 64abc ∫ ∫ ∫ r^3sin^3(f)cos^3(g)*r^3sin^3(f)sin^3(g)
0 0 0 +++^^^^^^^ ------- +++^^^^^^^ --------
*r^3cos^3(f)*r^2sin(f) drdfdg
+++^^^^^^^^ +++^^^^^^
2兀 兀 1
= 64abc( ∫sin^3(g)cos^3(g)dg )(∫ sin^7(f)cos^3(f)df )(∫r^11 dr)
0 0 0
問題一:我本來是把前兩個積分式變成 Beta function, 可是發現這樣好像是錯的,
請問有辦法用Beta做嗎?要如何修正?
問題二:我後來去用google看了sin^7(f)cos^3(f)的圖形
發現如果從 0 積到 兀 ,會是0耶???
圖:
https://www.google.com.tw/#q=(sin(x))%5E7(cos(x))%5E3
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