Question

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

Answer 1

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.