1. **Problem:** Evaluate the double integral $$\int_0^1 \int_1^3 xy \, dx \, dy$$. 2. **Formula and rules:** For double integrals over rectangular regions, integrate the inner integral first, treating the outer variable as constant. 3. **Step 1:** Integrate with respect to $x$: $$\int_1^3 xy \, dx = y \int_1^3 x \, dx = y \left[ \frac{x^2}{2} \right]_1^3 = y \left( \frac{9}{2} - \frac{1}{2} \right) = y \cdot 4 = 4y$$ 4. **Step 2:** Integrate the result with respect to $y$: $$\int_0^1 4y \, dy = 4 \int_0^1 y \, dy = 4 \left[ \frac{y^2}{2} \right]_0^1 = 4 \cdot \frac{1}{2} = 2$$ 5. **Answer:** The value of the integral is **2**.