Re: [方法] 100年成大第6題

[suhorng ( )]

※ 引述《kylestyle (最近有點煩)》之銘言：
: Find the average value of  f(x,y) = xy  over the quarter circle
: x^2 + y^2 ≦ 1  in the first quadrant
: 求板上大大解答

By definition, the average value is given by

_     1
f = -----∫∫f(x,y)dA
    A(Ω)  Ω

where Ω = {(x,y) | x^2+y^2≦1, x≧0, y≧0}
and A(Ω)  denotes  the  area  of Ω.  Thus

_    4   1 π/2
f = --- ∫∫r^2 cosθsinθ r dθ dr
     π  0 0

     4  r^4|1     cos 2θ |π/2
  = --- ---|  (- --------)|
     π  4 |0        4    |0

  = 1/2π

有錯請指正 m(_ _)m

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推 kylestyle:感謝大大 突然覺得這題頗簡單 = =
推 kylestyle:有一點搞不清楚 xy在換成極式相乘後
→ kylestyle:等於r^2 cosθsinθ r
→ kylestyle:這後面多出來的r是怎麼來的
→ suhorng:dx dy = dA = r dr dθ
→ suhorng:recall that dxdy=|∂(x,y)/∂(u,v)|dudv
推 kylestyle:多謝指點 全然懂了