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The parabolas y=x^2, y=x^2+1, y=(x-2)^2, y=(x-2)^2+1

The parabolas y=x^2, y=x^2+1, y=(x-2)^2, y=(x-2)^2+1 Intersectto form a curvilinear quadrilateral R. The change of variable u=y-x^2, v=y-(x-2)^2 map R onto a square in the uv-plane. Use the jacobian of the inverse transformation to compute the area of R.

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Check to make sure the question is correct. currently all parabolas are facing up, at least one should be reversed (-x or -x2), otherwise there is no upper bound, no shape and no area to compute. (1≤y<infinity)
Did you graph the four parabolas?
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1 Answer

 
The parabolas do all point up, but they also enclose a shape R that approximates a parallelogram with bounds in (x,y):    (3/4,25/16), (1,2), (1,1), and (5/4,25/16):
                      
 
From T (x,y) —> (u,v):  {y-x2, y-(x-2)2}
 
   The square is  u= y-x= 0   (for the boundary corresponding to y=x2)
                        u=y-(x2) = 1  (for the boundary corresponding to y=x2+1)
                        v = y-(x-2)2 = 0   (for the boundary corresponding to y=(x-2)2)
                        v = y-(x-2)2 = 1  (for the boundary corresponding to y=(x-2)2+1)
 
 
Solving T for x(u,v), y(u,v)  (i.e. writing x and y in terms of u and v):
                                  x = (1/4)(v-u+4)
                                  y = u + (1/16)(v-u+4)2
 
 
                 ∂x/∂u = -1/4
                 ∂x/∂v  = 1/4
                 ∂y/∂u  = 1+(1/16)(-2v+2u-8)
                 ∂y/∂v = (1/16)(2v-2u+8)
 
  Jx,y =  (∂x/∂u)(∂y/∂v) - (∂x/∂v)(∂y/∂u)
  Jx,y = -(1/4)[(1/16)(2v-2u+8)] - (1/4)[(1 + (1/16)(-2v+2u-8)] 
        = (-1/64)[2v-2u+8] - 1/4 - (1/64)[-2v+2u-8]        
        
   which reduces to -1/4,  so
 
                    | Jx,y | = 1/4   This is the area because we're dealing with a unit square in (u,v)
 
 
Check:
    Basically, we can integrate the "hard way" in (x,y) to check the result from above. 
 
    Integrate from x=3/4 to x=1   f(x) = (x2+1) - (x-2)
    and add the integral of  x=1 to x= 5/4  of  g(x) = (x-2)2+1- x2
 
    to get   [ 2-3 - (18/16 - 36/16) ]  + [ -50/16 + 100/16 - (-2 + 5) ]
             =    -1   +18/16  +  50/16 - 3
             =     -4  + 68/16
              =     4/16
              =  1/4   (looks ok)