※ 引述《iclaire (JOJO)》之銘言:
: http://ppt.cc/lVvX
: 分別是這2題
: 第(4)題我畫出範圍來
: 如果先積x的話 那是從x=-y-2到x=-y+2嗎@@
: 因為數字有點醜 所以想問問
: 第(1)題
: 目標函數應該是z
: 那限制式是只有橢圓方程式嗎?
: 想問該怎麼列式才好
: 謝謝!!
4. Evaluate ∫∫ x^2 - y^2 dxdy where is the region in the plane bounded by
R
the lines x-y = 0 , x-y = 1 , x+y = -2 and x+y = 2
sol: Let x - y = u , x + y = v
u v -u v
Then x = --- + --- , y = ---- + ---
2 2 2 2
| δx δx |
|----- -----|
| δu δv |
J = | |
| δy δy |
|----- -----|
| δu δv |
| 1 1 |
|--- ---|
| 2 2 | 1 1 1
= | | = --- + --- = ---
|-1 1 | 4 4 2
|--- ---|
| 2 2 |
1
|J| = ---
2
0 ≦ x - y ≦ 1 => 0 ≦ u ≦ 1
-2 ≦ x + y ≦ 2 => -2 ≦ v ≦ 2
∫∫ x^2 - y^2 dxdy
R^2
2 1 1
= ∫ ∫ (uv)(---) dudv
-2 0 2
2 u^2 |u=1
= ∫ (v)(-----)| dv
-2 4 |u=0
2 v v^2 |2
= ∫ --- dv = -----| = 0
-2 4 8 |-2
--
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