anonymous
  • anonymous
use the change of variables u=x+y, v=2x-3y to calculate the area inside the region bonded by the lines x+y=1, x+y=2, 2x-3y=2, 2x-3y=5.
OCW Scholar - Multivariable Calculus
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jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
Because it's a linear change of variables scaling factor remain constant i.e. dudv= 5dxdy we can even use jacobian to find out the scaling factor dudv= I J I dxdy J = det Ux Uy Vx Vy Here Ux = 1, Uy = 1, Vx = 2, Vy = -3 so J = det 1 1 2 -3 = -3-2 = -5 and I J I = I -5 I = 5 Now, Area = double integration dxdy = double integration 1/5dudv we have u = x+y, x+y = 1 & x+y=2, so u =1 & u =2 respectively same for v, so v=3 and v =5 In uv coordinate system region bounded by these lines is a rectangle of length 1 and width 3. so, Area = 1/5 double integration dudv = 1/5*area of the rectangle = 1/5*3 =3/5

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