Riemann Sum Area 9Fe72A

1. **State the problem:** We want to approximate the area under the curve $f(x) = \sqrt{x^2 + 3}$ from $x=2$ to $x=14$ using a Riemann sum with 6 equal subintervals and left endpoints. 2. **Determine the width of each subinterval:** The interval length is $14 - 2 = 12$. Dividing into 6 equal parts gives subinterval width: $$\Delta x = \frac{14 - 2}{6} = 2.$$ 3. **Identify the left endpoints:** Starting at $x=2$, the left endpoints for each subinterval are: $$x_0 = 2, \quad x_1 = 4, \quad x_2 = 6, \quad x_3 = 8, \quad x_4 = 10, \quad x_5 = 12.$$ 4. **Evaluate the function at each left endpoint:** $$f(x) = \sqrt{x^2 + 3}$$ Calculate each: $$f(2) = \sqrt{2^2 + 3} = \sqrt{4 + 3} = \sqrt{7}$$ $$f(4) = \sqrt{4^2 + 3} = \sqrt{16 + 3} = \sqrt{19}$$ $$f(6) = \sqrt{6^2 + 3} = \sqrt{36 + 3} = \sqrt{39}$$ $$f(8) = \sqrt{8^2 + 3} = \sqrt{64 + 3} = \sqrt{67}$$ $$f(10) = \sqrt{10^2 + 3} = \sqrt{100 + 3} = \sqrt{103}$$ $$f(12) = \sqrt{12^2 + 3} = \sqrt{144 + 3} = \sqrt{147}$$ 5. **Calculate the Riemann sum:** Using left endpoints, $$\text{Area} \approx \Delta x \sum_{i=0}^{5} f(x_i) = 2 \left( \sqrt{7} + \sqrt{19} + \sqrt{39} + \sqrt{67} + \sqrt{103} + \sqrt{147} \right).$$ 6. **Approximate the square roots:** $$\sqrt{7} \approx 2.6458, \quad \sqrt{19} \approx 4.3589, \quad \sqrt{39} \approx 6.2449,$$ $$\sqrt{67} \approx 8.1854, \quad \sqrt{103} \approx 10.1489, \quad \sqrt{147} \approx 12.1244.$$ 7. **Sum the values:** $$2.6458 + 4.3589 + 6.2449 + 8.1854 + 10.1489 + 12.1244 = 43.7083.$$ 8. **Multiply by $\Delta x = 2$ to get the approximate area:** $$\text{Area} \approx 2 \times 43.7083 = 87.4166.$$ **Final answer:** The approximate area under the curve using the left endpoint Riemann sum with 6 subintervals is about **87.42**.