作者suhorng ( )
看板trans_math
標題Re: [方法] 100年成大第6題
時間Sun Jun 24 17:59:07 2012
※ 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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◆ From: 59.115.146.209
1F：推 kylestyle:感謝大大 突然覺得這題頗簡單 = =125.228.152.142 06/24 18:36
2F：推 kylestyle:有一點搞不清楚 xy在換成極式相乘後125.228.152.142 06/24 18:41
3F：→ kylestyle:等於r^2 cosθsinθ r125.228.152.142 06/24 18:42
4F：→ kylestyle:這後面多出來的r是怎麼來的125.228.152.142 06/24 18:42
5F：→ suhorng:dx dy = dA = r dr dθ 59.115.146.209 06/24 18:45
6F：→ suhorng:recall that dxdy=|∂(x,y)/∂(u,v)|dudv 59.115.146.209 06/24 18:46
7F：推 kylestyle:多謝指點 全然懂了125.228.152.142 06/24 18:51