The surface area can be evaluated using the formula:
A = ∫∫√(1 + (∂z/∂x)^2 + (∂z/∂y)^2) dA
Where ∂z/∂x and ∂z/∂y are the partial derivatives of z with respect to x and y, respectively.
First, we find the partial derivatives:
∂z/∂x = 2
∂z/∂y = 2
Then, we plug these values into the formula:
A = ∫∫√(1 + 2^2 + 2^2) dA
A = ∫∫√(1 + 4 + 4) dA
A = ∫∫√(9) dA
A = ∫∫3 dA
Now, we integrate with respect to x and y:
A = ∫[0,1]∫[0,1] 3 dy dx
A = 3∫[0,1]y∣[0,1] dx
A = 3[y]∣[0,1]
A = 3(1 - 0)
A = 3
So, the area of the surface z=1+2x+2y for 0<=x<=1 and 0 <= y <= 1 is 3.