1. **State the problem:** Evaluate the double integral $$\iint_R (x + y) \, dA$$ where the region $$R$$ is the rectangle defined by $$0 \leq x \leq 2$$ and $$1 \leq y \leq 3$$. 2. **Set up the integral:** Since $$R$$ is a rectangle, the double integral can be written as an iterated integral: $$\int_{y=1}^{3} \int_{x=0}^{2} (x + y) \, dx \, dy$$ 3. **Integrate with respect to $$x$$ first:** $$\int_0^2 (x + y) \, dx = \int_0^2 x \, dx + \int_0^2 y \, dx = \left[ \frac{x^2}{2} \right]_0^2 + y \left[ x \right]_0^2 = \frac{2^2}{2} + y(2) = 2 + 2y$$ 4. **Now integrate with respect to $$y$$:** $$\int_1^3 (2 + 2y) \, dy = \int_1^3 2 \, dy + \int_1^3 2y \, dy = 2[y]_1^3 + 2 \left[ \frac{y^2}{2} \right]_1^3 = 2(3 - 1) + (3^2 - 1^2) = 4 + (9 - 1) = 4 + 8 = 12$$ 5. **Final answer:** The value of the double integral is $$12$$.