Question

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.

Answer 1

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