Let $C$ be a simple closed curve in $\mathbf{R^2}$ with positive orientation enclosing a region $D$. Assume that D has area equal to 2 and centroid $(\bar{x}, \bar{y})=(3,4)$. Let $\mathbf{F}= \langle y^2,x^2+3x\rangle$. Calculate the value of $\int_C \mathbf{F}\cdot d\mathbf{r}$.
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