Question

Using Green's theorem,evaluate ∮c(x+y)dx+ x²dy, C is the triangle

with vertices at (0,0). (2,0) and (2, 4), taken in that order.

Answer 1

The components of the vector field are

$P ( x , y ) = x^2 , Q ( x , y ) = x + y$

Using the Green’s formula

$∬_ R (\frac{ ∂ Q}{ ∂ x} − \frac{∂ P}{ ∂ y} ) d x d y = ∮_ C P d x + Q d y$

we transform the line integral into the double integral:

$I = ∮_ C x^2 d y + ( x + y ) d x = ∬_ R ( \frac{∂ ( x + y )}{ ∂ x} − \frac{∂ ( x y )}{ ∂ y} ) d x d y = ∬_ R ( 2x − 1 ) d x d y$

$\int _0^2\int _0^{2x}\left(2x-1\right)dydx\\ =\int _0^22x\left(2x-1\right)dx\\ =\frac{20}{3}$