Calculate the Green’s value for the functions F = y2 and G = x2 for the region x = 1 and y = 2 from origin.

To use Green’s theorem to calculate the value for the given functions (F = y^2) and (G = x^2) across a specified region, we first need to understand the theorem in the context of a region (R) and its positively oriented boundary (C). The theorem states:

[oint_C (L dx + M dy) = int int_R left(frac{partial M}{partial x} – frac{partial L}{partial y}right) dA]

where (L) and (M) are the components of a vector field, that is, (mathbf{F} = Lmathbf{i} + Mmathbf{j}).

For the given functions, if we interpret (F = y^2) as (L) and (G = x^2) as (M), then we have:

– (L = F = y^2)

– (M = G = x^2)

To apply Green’s theorem, we need to evaluate (frac{partial M}{partial x} – frac{partial L}{partial y}):

[

frac{partial M}{partial x} = frac{partial (x^2)}{partial x} = 2x

]

[

frac{partial L}{partial y} = frac{partial (y^2)}{partial y} =