1. **Problem Statement:** Estimate the area under the curve of the function $f$ given by the table: | $x$ | 0 | 1 | 2 | 3 | |-----|---|---|---|---| | $f(x)$ | 2 | 3 | 5 | 4 | using (a) Left sum, (b) Right sum, and (c) Midpoint sum with $n=3$ subintervals from $x=0$ to $x=3$. 2. **Formula and Explanation:** - The interval length is $3 - 0 = 3$. - Number of subintervals $n=3$, so each subinterval width is $\Delta x = \frac{3}{3} = 1$. - **Left sum:** Use the left endpoint of each subinterval to find the height. - **Right sum:** Use the right endpoint of each subinterval. - **Midpoint sum:** Use the midpoint of each subinterval. 3. **Calculations:** (a) Left sum: $$\text{Left sum} = \Delta x \times [f(x_0) + f(x_1) + f(x_2)] = 1 \times (2 + 3 + 5) = 10$$ (b) Right sum: $$\text{Right sum} = \Delta x \times [f(x_1) + f(x_2) + f(x_3)] = 1 \times (3 + 5 + 4) = 12$$ (c) Midpoint sum: - Midpoints are $x=0.5, 1.5, 2.5$. - We estimate $f(0.5)$, $f(1.5)$, and $f(2.5)$ by averaging adjacent values: - $f(0.5) \approx \frac{f(0) + f(1)}{2} = \frac{2 + 3}{2} = 2.5$ - $f(1.5) \approx \frac{f(1) + f(2)}{2} = \frac{3 + 5}{2} = 4$ - $f(2.5) \approx \frac{f(2) + f(3)}{2} = \frac{5 + 4}{2} = 4.5$ $$\text{Midpoint sum} = \Delta x \times [2.5 + 4 + 4.5] = 1 \times 11 = 11$$ 4. **Final answers:** - Left sum estimate: $10$ - Right sum estimate: $12$ - Midpoint sum estimate: $11$