1. **State the problem:** Evaluate the integral $$\frac{1}{2} \int_{-1}^{1} \sqrt{1 - x^2} \, dx$$. 2. **Recognize the integral:** The integral $$\int_{-1}^{1} \sqrt{1 - x^2} \, dx$$ represents the area of a semicircle of radius 1 centered at the origin on the x-axis. 3. **Formula for the area of a circle:** The area of a full circle with radius $r$ is $$\pi r^2$$. 4. **Area of the semicircle:** Since the integral covers from $-1$ to $1$, it corresponds to the upper half of the circle, so the area is $$\frac{\pi r^2}{2} = \frac{\pi \cdot 1^2}{2} = \frac{\pi}{2}$$. 5. **Multiply by the factor outside the integral:** The original expression has a factor of $\frac{1}{2}$, so multiply the semicircle area by $\frac{1}{2}$: $$\frac{1}{2} \times \frac{\pi}{2} = \frac{\pi}{4}$$. 6. **Final answer:** $$\boxed{\frac{\pi}{4}}$$