Rewrite the integral: ∬ (x - 3y)dA where R is the triangular region with vertices (0, 0), (2, 1), (1, 2) in the xy-plane using the transformation x = 2u + v, y = u + 2v.
(a) Find the image of R.
(i) Find the image of the side S1 joining the vertices (0,0), (2,1):
v = 1-u, for 0 ≤ u ≤ 1
(ii) Find the image of the side S2 joining the vertices (0,0), (1,2):
u = v, for 0 ≤ v ≤ 1
(iii) Find the image of the side S3 joining the vertices (2,1), (1,2):
v = 1-u, for 0 ≤ u ≤ 1
(b) Find the Jacobian of the transformation (with alphabetical ordering):
(c) Write the transformed integral: ∫∫ dvdu,
where a =
b =
c = , and
d = .
(d) Evaluate the integral: