To solve this problem, we’ll use the upper and lower sums (also called Riemann sums) to approximate the area under the curve given by $y = \sqrt{1 - x^2}$.

This curve is actually the top half of a circle with radius 1, centered at the origin, so the region under this curve from $x = -1$ to $x = 1$ represents a semicircle with area $\frac{\pi}{2} \approx 1.5708$. However, to approximate this area using Riemann sums, we need to apply the method specified.

Step-by-Step Solution Outline

1. Define the Interval and Function: The interval is typically from $x = -1$ to $x = 1$ (as this is the range for a semicircle). The function to evaluate is: $f(x) = \sqrt{1 - x^2}$

2. Determine Subinterval Width: Let’s denote the number of subintervals as $n$. Since the problem doesn’t specify the number of subintervals, please confirm this information if available. Otherwise, we can proceed with a general calculation for arbitrary $n$ or assume a standard value (such as $n = 4$).

   The width of each subinterval $\Delta x$ is: $\Delta x = \frac{1 - (-1)}{n} = \frac{2}{n}$

3. Calculate Upper and Lower Sums: For the upper and lower sums, we evaluate $f(x)$ at the endpoints of each subinterval to find the maximum (for the upper sum) and minimum (for the lower sum) function values within each subinterval.

4. Approximate the Sums:

   • Upper Sum: Sum the areas of rectangles where the height is determined by the maximum value of $f(x)$ in each subinterval.
   • Lower Sum: Sum the areas of rectangles where the height is determined by the minimum value of $f(x)$ in each subinterval.

5. Round Answers: Round the upper and lower sums to three decimal places, as specified.

Could you confirm the number of subintervals $n$? Once confirmed, I’ll proceed with the calculations.