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Integration (of a flat area) and parameterization

1. The function f(x,y) = x + y

2. The area A is formed by the lines : x = 0 and x = pi/4 and by the graphs : x + cos(x) and x + sin(x)

3. I have to parametrize A

Could this be a parameterization of A :

r(u,v) = (u,v*f(u)+(1-v)*g(u))

assuming f(u) = u+ cos(u) and g(u) = u + sin(u)

r(u,v) = (u, v*(u+cos(u))+(1-v)*(u+sin(u))) <->

r(u,v) = (u, v*cos(u)+u+sin(u)-v*sin(u)) and

Attachment 27437

? ? ?

I would really appreciate it if someone could look at this and give me some feedback because I'm not sure if the parametrization is correct :/ and I cannot solve the rest of the problem if the parametrization is incorrect.