Question

Use Green's Theorem to evaluate the line integral. $$ \begin{aligned} &\int_{C} 2 x y d x+(x+y) d y\\\ &C: \text { boundary of the region lying between the graphs of } y=0 \text { and }\\\ &y=4-x^{2} \end{aligned} $$

Short answer

The value of the line integral is -\frac{32}{5}

Step-by-step solution

Construct the Vector Field

Firstly, the given integral needs to be represented as a line integral of a vector field. The given integral can be rewritten as \[ \int_C x \cdot dx + y \cdot dy \] So, the vector field \( F \) is \( F(x, y) = (2xy, x+y) \)

Apply Green's Theorem

Green's Theorem states that \( \int_C F \cdot dr = \iint_D (\frac{\partial{Q}}{\partial{x}} - \frac{\partial{P}}{\partial{y}}) dA \). The derivative \( \frac{\partial{Q}}{\partial{x}} = 1 \) and \( \frac{\partial{P}}{\partial{y}} = 2x \)

Evaluate the Double Integral

The region of integration D is defined by the curves \( y = 0 \) and \( y = 4 - x^2 \). The limits of the x-integral are \( x = -2 \) to \( x = 2 \). The limits of the y-integral are \( y = 0 \) to \( y = 4 - x^2 \). Therefore, the double integral can be written as \[ \int_{-2}^2 ( \int_{0}^{4 - x^2} (1 - 2x) dy) dx = \int_{-2}^{2} [(1 - 2x)(4-x^2)] dx = -\frac{32}{5} \]