Use the given transformation to evaluate the integral.
(x − 8y) dA, | |
R |
where R is the triangular region with vertices (0, 0), (7, 1), and (1, 7).
x = 7u + v, y = u + 7v
Solution :- Given that x = 7u + v and y = u + 7v.
Then, the boundary lines of R transform as we get
x = 7y so v = 0
also y = 7x so u = 0
and y = -x + 8 so u + v = 1.
Now the Jacobian ∂(x,y)/∂(u,v) equals
|7 1|
|1 7| = 48.
So change of variables we get
∫∫R (x - 8y) dA
= ∫(v = 0 to 1) ∫(u = 0 to 1-v) [(7u+v) - 8(u+7v)] * 48 du dv
= ∫(v = 0 to 1) ∫(u = 0 to 1-v) -48(u + 55v) du dv
= ∫(v = 0 to 1) -48((1/2)u^2 + 55uv) {for u = 0 to 1-v} dv
= ∫(v = 0 to 1) -24((1 - v)^2 + 110(1 - v)v) dv
= ∫(w = 1 to 0) -24(w^2 + 110w(1 - w)) * -dw, letting w = 1 - v
= ∫(w = 0 to 1) -24(110w - 109w^2) dw
= -24(55 - 109/3)
= -448.
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