1. The problem asks to calculate three types of Riemann sums for the function $g(x)$ on given intervals: A. Right Riemann sum $R_4$ on $[0,2]$ B. Left Riemann sum $L_3$ on $[1,4]$ C. Midpoint Riemann sum $M_3$ on $[0.5,3.5]$ 2. The formulas for Riemann sums are: - Right sum: $$R_n = \sum_{i=1}^n f(x_i) \Delta x$$ where $x_i$ are the right endpoints. - Left sum: $$L_n = \sum_{i=0}^{n-1} f(x_i) \Delta x$$ where $x_i$ are the left endpoints. - Midpoint sum: $$M_n = \sum_{i=0}^{n-1} f\left(\frac{x_i + x_{i+1}}{2}\right) \Delta x$$ where midpoints are used. 3. We need to find $\Delta x$ and the sample points for each sum. A. For $R_4$ on $[0,2]$: - $\Delta x = \frac{2-0}{4} = 0.5$ - Right endpoints: $x_1=0.5$, $x_2=1.0$, $x_3=1.5$, $x_4=2.0$ - Using the graph description, approximate $g(0.5)$, $g(1.0)$, $g(1.5)$, $g(2.0)$: - $g(0.5) \approx 0.5$ - $g(1.0) \approx 1.2$ - $g(1.5) \approx 1.8$ - $g(2.0) \approx 2.1$ - Calculate: $$R_4 = 0.5 \times (0.5 + 1.2 + 1.8 + 2.1) = 0.5 \times 5.6 = 2.8$$ B. For $L_3$ on $[1,4]$: - $\Delta x = \frac{4-1}{3} = 1$ - Left endpoints: $x_0=1$, $x_1=2$, $x_2=3$ - Approximate $g(1)$, $g(2)$, $g(3)$: - $g(1) \approx 1.2$ - $g(2) \approx 2.1$ - $g(3) \approx 1.8$ - Calculate: $$L_3 = 1 \times (1.2 + 2.1 + 1.8) = 5.1$$ C. For $M_3$ on $[0.5,3.5]$: - $\Delta x = \frac{3.5-0.5}{3} = 1$ - Subintervals: $[0.5,1.5]$, $[1.5,2.5]$, $[2.5,3.5]$ - Midpoints: - $1.0$, $2.0$, $3.0$ - Approximate $g(1.0)$, $g(2.0)$, $g(3.0)$: - $g(1.0) \approx 1.2$ - $g(2.0) \approx 2.1$ - $g(3.0) \approx 1.8$ - Calculate: $$M_3 = 1 \times (1.2 + 2.1 + 1.8) = 5.1$$ 4. Final answers: - $R_4 = 2.8$ - $L_3 = 5.1$ - $M_3 = 5.1$