1. **Problem Statement:** Evaluate the double integral $$\int_0^1 \int_0^x y \, dy \, dx$$. 2. **Formula and Explanation:** The integral is over the region where $y$ goes from 0 to $x$, and $x$ goes from 0 to 1. The function to integrate is $y$. 3. **Step 1: Inner integral with respect to $y$:** $$\int_0^x y \, dy = \left[ \frac{y^2}{2} \right]_0^x = \frac{x^2}{2}$$ 4. **Step 2: Outer integral with respect to $x$:** $$\int_0^1 \frac{x^2}{2} \, dx = \frac{1}{2} \int_0^1 x^2 \, dx = \frac{1}{2} \left[ \frac{x^3}{3} \right]_0^1 = \frac{1}{2} \cdot \frac{1}{3} = \frac{1}{6}$$ 5. **Final answer:** $$\boxed{\frac{1}{6}}$$ This means the value of the double integral is $\frac{1}{6}$.