求曲線圍成的面積(2)

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圖片參考：http://img20.imageshack.us/img20/8406/17140746.png


2010-10-03 12:30:37 補充：
http://img20.imageshack.us/img20/8406/17140746.png

2010-10-03 14:17:28 補充：
對,那上限應該是sqrt(2),是錯手打錯了.
但面積好像沒有sqrt(2),讓我再對一下.

2010-10-03 14:24:32 補充：
Integrate v du 的上限是sqrt(2),轉化t後上下限沒錯了.

To: Nelson
u^4+v^4=4面積的上下限不是0~1

2010-10-03 14:12:57 補充：
謹供參考:坐標變換(u,v)=(x-y, 2x+3y), 則Jacobian= 1/5
面積=∫∫_D dxdy=∫∫_D (1/5)dudv (D: u^4+v^4=4)
=(4/5)∫[0~√2]∫[0~(4-u^4)^(1/4)] dudv [u^2= 2sint ]
=(4/5)∫[0~π/2] (cost)^(3/2)/(√sint) dt
=(2/5)B(5/4, 1/4)=(2/5)Γ(5/4)Γ(1/4)/Γ(3/2)
=(2/5)*(1/4)[Γ(1/4)]^2/[(1/2)Γ(1/2)]=[Γ(1/4)]^2/(5√π)