1. **Problem:** Evaluate the double integral $$\int_0^2 \int_0^1 4xy \, dx \, dy$$. 2. **Formula and rules:** For double integrals, integrate with respect to the inner variable first, then the outer variable. 3. **Step 1:** Integrate with respect to $x$: $$\int_0^1 4xy \, dx = 4y \int_0^1 x \, dx = 4y \left[ \frac{x^2}{2} \right]_0^1 = 4y \cdot \frac{1}{2} = 2y$$ 4. **Step 2:** Now integrate with respect to $y$: $$\int_0^2 2y \, dy = 2 \int_0^2 y \, dy = 2 \left[ \frac{y^2}{2} \right]_0^2 = \cancel{2} \cdot \frac{4}{\cancel{2}} = 4$$ 5. **Answer:** The value of the integral is **4**.