1. **Problem Statement:** We want to approximate the area of the region bounded by the graph of $f(x) = \sin(x)$ and the x-axis between $x=0$ and $x=0.5$ using a Riemann sum. 2. **Formula and Explanation:** A Riemann sum approximates the area under a curve by dividing the interval into smaller subintervals, calculating the function value at specific points in these subintervals, and summing the areas of rectangles formed. The Riemann sum is given by: $$ S_n = \sum_{i=1}^n f(x_i^*) \Delta x $$ where $\Delta x = \frac{b-a}{n}$ is the width of each subinterval, and $x_i^*$ is a sample point in the $i$-th subinterval. 3. **Choosing Parameters:** - Interval: $[0, 0.5]$ - Number of subintervals: Let's choose $n=5$ for simplicity. - Width of each subinterval: $$ \Delta x = \frac{0.5 - 0}{5} = 0.1 $$ 4. **Sample Points:** Using right endpoints for the Riemann sum: $$ x_i^* = 0 + i \times 0.1, \quad i=1,2,3,4,5 $$ So the points are $0.1, 0.2, 0.3, 0.4, 0.5$. 5. **Calculate Function Values:** $$ f(0.1) = \sin(0.1) \approx 0.0998 $$ $$ f(0.2) = \sin(0.2) \approx 0.1987 $$ $$ f(0.3) = \sin(0.3) \approx 0.2955 $$ $$ f(0.4) = \sin(0.4) \approx 0.3894 $$ $$ f(0.5) = \sin(0.5) \approx 0.4794 $$ 6. **Compute the Riemann Sum:** $$ S_5 = 0.1 \times (0.0998 + 0.1987 + 0.2955 + 0.3894 + 0.4794) $$ $$ S_5 = 0.1 \times 1.4628 = 0.14628 $$ 7. **Interpretation:** The approximate area under $f(x) = \sin(x)$ from $0$ to $0.5$ using a right-endpoint Riemann sum with 5 subintervals is about $0.1463$. This method can be improved by increasing $n$ or using other sample points (left endpoints, midpoints).