Questions: Use Green's Theorem to evaluate the line integral [ intC(y-x) d x+(2 x-y) d y ] for the given path. [ C: x=2 cos (theta), y=sin (theta), 0 leq theta leq 2 pi ]

To evaluate the line integral using Green's Theorem, we first need to identify the vector field \(\mathbf{F} = (y-x, 2x-y)\). Green's Theorem relates a line integral around a simple closed curve \(C\) to a double integral over the region \(R\) enclosed by \(C\). The theorem states:

\[ \oint_{C} \mathbf{F} \cdot d\mathbf{r} = \iint_{R} \left( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \right) \, dA \]

where \(\mathbf{F} = (M, N)\). For this problem, \(M = y-x\) and \(N = 2x-y\). We need to compute the partial derivatives \(\frac{\partial N}{\partial x}\) and \(\frac{\partial M}{\partial y}\), and then evaluate the double integral over the region \(R\) enclosed by the ellipse described by the parametric equations \(x = 2\cos(\theta)\) and \(y = \sin(\theta)\).

Para resolver la integral de línea usando el Teorema de Green, seguiremos los pasos detallados a continuación.

La integral de línea dada es: \[ \int_{C}(y-x) \, dx + (2x-y) \, dy \] Podemos identificar la función vectorial \(\mathbf{F} = (P, Q)\) donde \(P = y-x\) y \(Q = 2x-y\).

El Teorema de Green establece que para una curva cerrada \(C\) y una región \(R\) que encierra, la integral de línea es igual a la integral doble sobre \(R\) de la derivada cruzada: \[ \oint_{C} P \, dx + Q \, dy = \iint_{R} \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \, dA \]

Calculemos las derivadas parciales necesarias: \[ \frac{\partial Q}{\partial x} = \frac{\partial}{\partial x}(2x-y) = 2 \] \[ \frac{\partial P}{\partial y} = \frac{\partial}{\partial y}(y-x) = 1 \]

Sustituyendo en la integral doble: \[ \iint_{R} \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \, dA = \iint_{R} (2 - 1) \, dA = \iint_{R} 1 \, dA \] Esto significa que la integral doble es simplemente el área de la región \(R\).

La curva \(C\) está dada por \(x = 2 \cos(\theta)\) y \(y = \sin(\theta)\), que describe una elipse. La ecuación paramétrica corresponde a una elipse con semieje mayor \(a = 2\) y semieje menor \(b = 1\).

El área de una elipse es: \[ A = \pi \cdot a \cdot b = \pi \cdot 2 \cdot 1 = 2\pi \]

Por lo tanto, la integral de línea evaluada usando el Teorema de Green es: \[ \boxed{2\pi} \]