[微積] 請教高手 double integrals in polar coordinates

[vo90230014 (vo)]

在G.zill工數中9.11double integral in polar coordinates EX3如下

find the volume of the solid that is under the hemisphere
 z=sqrt(1-x^2-y^2) and above the region bounded by the graph
 of the circle x^2+y^2-y=0

此題作者用polar coordinate轉換去做

    π/2  sinθ      1/2
= 2∫   ∫    (1-r^2)    r drdθ=.....=π/3-4/9
     0    0



 但問題來了 作者在最後remarks

However, carry out the iterated integration

  π sinθ       1/2
=∫ ∫    (1-r^2)   r drdθ=π/3
  0  0

and see if the incorrect π/3. What goes wrong ?

為何明明是對稱圖形卻不可以這麼做ㄋ?
我卡在這裡好久了請各位高手幫幫我

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