1. **Problem:** Calculate the double integral $$\int_0^x \int_0^x \sin y \, dy \, dx$$. 2. **Formula and rules:** The double integral over a rectangular region can be computed by iterated integrals. Here, the inner integral is with respect to $y$, and the outer integral with respect to $x$. Since the limits are constants or depend on $x$, we integrate step-by-step. 3. **Step 1: Compute the inner integral** $$\int_0^x \sin y \, dy = [-\cos y]_0^x = -\cos x + 1$$ 4. **Step 2: Substitute inner integral result into outer integral** $$\int_0^x (-\cos x + 1) \, dx = \int_0^x (1 - \cos x) \, dx$$ 5. **Step 3: Integrate outer integral** $$\int_0^x 1 \, dx - \int_0^x \cos x \, dx = [x]_0^x - [\sin x]_0^x = x - \sin x$$ 6. **Answer:** The value of the double integral is $$x - \sin x$$. This completes the solution for the first problem in the list.