How to set this integral?

Question

Let R be the region bounded by the curves y=x/2, y=3x, y=1/x and y=4/x. The change of variables are u=y/x and v=xy. Evaluate the integral ydA. I understand that you have to change the bounds of x and y to u and v. I just don't know how.

AfOreUgNy · Accepted Answer

When   u = y/x   and   v = xy    the  region  is bounded by :
 u = 1/2   ;   u = 3   ;   v = 1   ;   v = 4  .

 (from  y = x/2  ==>  y/x = 1/2  ==>   u = 1/2   and  so  on ...)

 Now find  x = x(u,v)  and  y = y(u,v) :
 x = √(v/u)
 y = √(uv)

 Next find
 ∂x/∂u =  -1/2 √(v/u³)
 ∂x/∂v = 1 / 2√(uv)
 ∂y/∂u  = v / 2√(uv)
 ∂y/∂v  = u / 2√(uv)

 Changing variables , the Jacobian matrix' determinant is 
 | ∂(x,y)/∂(u,v) | = 1 / 2u

 Then the area element
 dA = dx dy = 1/2u du dv

 The integral  ∫ y dA  becomes

 1/2 ∫(v = 1 to 4) ∫(u = 1/2 to 3)  √(v/u) du dv = 
 = 1/2 (2√3 - √2) ∫(v = 1 to 4) √v dv = 
 = 7/3 (2√3 - √2)

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How to set this integral?

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