# 高等微積分......急急急

let Ｓ be the region in the first quadrant bounded by the curves xy＝１,xy＝４,and lines y＝x,y＝４x. find the area and the centriod of Ｓ using the transmation u＝xy,v＝y/x.

### 3 個解答

• 1 0 年前
最佳解答

1. the area of Ｓ= double integral [over S] {1} dA; call it A(S)

2. the the centriod of Ｓ=(x0, y0), where x0 = 1/(A(S)) * double integral [over S] {x} dA; y0 = 1/(A(S)) * double integral [over S] {y} dA.

以上是你要的[被積分函數 是什麼]

Using the transmation u＝xy,v＝y/x => = x=(u/v)^(1/2); y=(uv)^(1/2) => x_u= (1/2) * u^(-1/2) * v^(-3/2); x_v = (-1/2) * u^(1/2) * v^(-5/2);

y_u= (1/2) * u^(-1/2) * v^(1/2); x_v = (1/2) * u^(1/2) * v^(-1/2); => The transformation Jacobian |J|=(1/2)*v^(-2) ----大概是大大說的九摳逼吧?

Thus dA=dxdy= |J| dudv and S--->R after the change of variables.

Calculating A(S): (x,y)-->(u,v) A(S) = double integral [over R] {1* (1/2)*v^(-2) } dudv, where R is the rectangle {(u,v)| 1<=u<=4, 1<=v<=4} on uv-plane. ------ => A(S)=9/8;

Using formulas in 2, you should be able to finish finding the centroid.

• 1 0 年前

利用Jacobian變換之後的圖形較易算，自變數改為u，v，積分上下限也都是介於1和4之間

• linch
Lv 7
1 0 年前

被積分函數就是 1 啊!