Answer
The largest region is that bounded by the circle \(x^2 + y^2 = 1\).
Work Step by Step
Step 1: Given \[ \oint_C \left( \frac{1}{3}y^3 + \left(x - \frac{1}{3}x^3\right) \right) \, dy \] Step 2: Here \[ \begin{align*} \mathbf{F}(x, y) &= \left(\frac{1}{3}y^3, x - \frac{1}{3}x^3\right) \\ f(x, y) &= \frac{1}{3}y^3 \\ g(x, y) &= x - \frac{1}{3}x^3 \end{align*} \] Then by using Green’s Theorem \[ \begin{align*} \oint_C \mathbf{F} \cdot d\mathbf{r} + \iint_R \left(\frac{\partial g}{\partial x} - \frac{\partial f}{\partial y}\right) \, dA &= \iint_R \left(1 - x^2 - y^2\right) \, dA \end{align*} \] where \(R\) is the region enclosed by \(C\). The value of this integral is maximum if the integration extends over the largest region for which the integrand \(\frac{1}{3}y^3 + \left(x - \frac{1}{3}x^3\right) = 1 - x^2 - y^2\) is nonnegative. So we want \[ \begin{align*} 1 - x^2 - y^2 &\geq 0 \\ x^2 + y^2 &\leq 1 \end{align*} \] Hence, the largest region is that bounded by the circle \[ x^2 + y^2 = 1 \] Result: The largest region is that bounded by the circle \(x^2 + y^2 = 1\).
