1. **State the problem:** We need to evaluate the line integral $$\int_C \mathbf{F} \cdot d\mathbf{r}$$ where $$\mathbf{F}(x,y) = \left(8 - 14xy^2 + 2y e^{2x}\right) \mathbf{i} + \left(e^{2x} - 14x^2 y\right) \mathbf{j}$$ and $$C$$ is a curve from $$(-1, 2)$$ to $$(0, 0)$$. 2. **Check if $$\mathbf{F}$$ is conservative:** A vector field $$\mathbf{F} = P\mathbf{i} + Q\mathbf{j}$$ is conservative if $$\frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x}$$. Given: $$P = 8 - 14xy^2 + 2y e^{2x}$$ $$Q = e^{2x} - 14x^2 y$$ Calculate partial derivatives: $$\frac{\partial P}{\partial y} = -14x \cdot 2y + 2 e^{2x} = -28xy + 2 e^{2x}$$ $$\frac{\partial Q}{\partial x} = 2 e^{2x} - 14 \cdot 2x y = 2 e^{2x} - 28xy$$ Since $$\frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x} = -28xy + 2 e^{2x}$$, $$\mathbf{F}$$ is conservative. 3. **Find potential function $$\phi(x,y)$$:** Since $$\mathbf{F} = \nabla \phi$$, we have: $$\frac{\partial \phi}{\partial x} = P = 8 - 14xy^2 + 2y e^{2x}$$ $$\frac{\partial \phi}{\partial y} = Q = e^{2x} - 14x^2 y$$ Integrate $$P$$ with respect to $$x$$: $$\phi(x,y) = \int (8 - 14xy^2 + 2y e^{2x}) dx = 8x - 14y^2 \int x dx + 2y \int e^{2x} dx + h(y)$$ Calculate each integral: $$\int x dx = \frac{x^2}{2}$$ $$\int e^{2x} dx = \frac{e^{2x}}{2}$$ So: $$\phi(x,y) = 8x - 14y^2 \cdot \frac{x^2}{2} + 2y \cdot \frac{e^{2x}}{2} + h(y) = 8x - 7x^2 y^2 + y e^{2x} + h(y)$$ 4. **Find $$h(y)$$ by differentiating $$\phi$$ with respect to $$y$$ and equate to $$Q$$:** $$\frac{\partial \phi}{\partial y} = -14x^2 y + e^{2x} + h'(y)$$ Set equal to $$Q$$: $$-14x^2 y + e^{2x} + h'(y) = e^{2x} - 14x^2 y$$ Simplify: $$h'(y) = 0$$ So $$h(y)$$ is a constant, which we can take as zero. 5. **Evaluate the line integral using the potential function:** Since $$\mathbf{F}$$ is conservative, the line integral depends only on endpoints: $$\int_C \mathbf{F} \cdot d\mathbf{r} = \phi(0,0) - \phi(-1,2)$$ Calculate: $$\phi(0,0) = 8 \cdot 0 - 7 \cdot 0^2 \cdot 0^2 + 0 \cdot e^{0} = 0$$ $$\phi(-1,2) = 8(-1) - 7(-1)^2 (2)^2 + 2 e^{2(-1)} = -8 - 7 \cdot 1 \cdot 4 + 2 e^{-2} = -8 - 28 + 2 e^{-2} = -36 + 2 e^{-2}$$ 6. **Final answer:** $$\int_C \mathbf{F} \cdot d\mathbf{r} = 0 - (-36 + 2 e^{-2}) = 36 - 2 e^{-2}$$