Questions: Use Green's Theorem to evaluate the line integral. Assume the curve is oriented counterclockwise. ∮(2x+sinh y) dy-(5y^2+e^(x^2)) dx, where C is the boundary of the square with vertices (1,3),(3,3),(3,5), and (1,5). ∮(2x+sinh y) dy-(5y^2+e^(x^2)) dx= (Type an exact answer.)

Solution Steps

To evaluate the line integral using Green's Theorem, we need to convert the line integral over the closed curve \( C \) into a double integral over the region \( R \) enclosed by \( C \). Green's Theorem states that for a vector field \(\mathbf{F} = (M, N)\), the line integral \(\oint_C M \, dx + N \, dy\) is equal to the double integral \(\iint_R \left( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \right) \, dA\). Here, \( M = -(5y^2 + e^{x^2}) \) and \( N = 2x + \sinh y \). We will compute the partial derivatives, set up the double integral over the square region, and evaluate it.

We start with the vector field defined by the line integral: \[ \oint_{C} (2x + \sinh y) \, dy - (5y^2 + e^{x^2}) \, dx \] Here, we identify \( M = -(5y^2 + e^{x^2}) \) and \( N = 2x + \sinh y \).

Next, we compute the partial derivatives: \[ \frac{\partial N}{\partial x} = 2 \] \[ \frac{\partial M}{\partial y} = -10y \]

Using Green's Theorem, we find the integrand for the double integral: \[ \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} = 2 + 10y \]

The region \( R \) is defined by the square with vertices \((1,3)\), \((3,3)\), \((3,5)\), and \((1,5)\). The limits of integration are: \[ x \text{ from } 1 \text{ to } 3, \quad y \text{ from } 3 \text{ to } 5 \]

We evaluate the double integral: \[ \iint_R (2 + 10y) \, dA = \int_{1}^{3} \int_{3}^{5} (2 + 10y) \, dy \, dx \] Calculating this integral gives us the result: \[ \text{Result} = 168 \]

Final Answer