Let's carefully work through each part of the problem step-by-step. --- ### **Problem Overview:** Given the function: \[ f(x) = 2 \sin x, \quad 0 \leq x \leq \frac{3\pi}{2} \] We are asked to compute **Riemann sums with 6 subintervals**: - **Part (a):** Using **right endpoints**. - **Part (b):** Using **midpoints**. --- ### **Part (a): Riemann Sum with Right Endpoints (R6)** --- ### **Step 1: Determine the subintervals** Total interval length: \[ b - a = \frac{3\pi}{2} - 0 = \frac{3\pi}{2} \] Number of subintervals: \[ n = 6 \] Width of each subinterval: \[ \Delta x = \frac{b - a}{n} = \frac{\frac{3\pi}{2}}{6} = \frac{3\pi/2}{6} = \frac{3\pi}{2} \times \frac{1}{6} = \frac{3\pi}{12} = \frac{\pi}{4} \] So: \[ \boxed{ \Delta x = \frac{\pi}{4} } \] --- ### **Step 2: Determine the right endpoints** The subintervals are: \[ [x_0, x_1], [x_1, x_2], \dots, [x_5, x_6] \] where: \[ x_0 = 0 \] and \[ x_k = a + k \Delta x = 0 + k \times \frac{\pi}{4} \] for \(k=0,1,2,3,4,5,6\). The right endpoints are at \(x_k\) for \(k=1\) to \(6\): \[ x_1 = \frac{\pi}{4}, \quad x_2 = \frac{2\pi}{4} = \frac{\pi}{2}, \quad x_3 = \frac{3\pi}{4}, \quad x_4 = \pi, \quad x_5 = \frac{5\pi}{4}, \quad x_6 = \frac{6\pi}{4} = \frac{3\pi}{2} \] --- ### **Step 3: Compute the Riemann sum (right endpoints)** \[ R_6 = \sum_{k=1}^6 f(x_k) \Delta x \] Calculate each \(f(x_k)\): \[ f(x) = 2 \sin x \] Evaluate: - \(f(x_1) = 2 \sin \left(\frac{\pi}{4}\right) = 2 \times \frac{\sqrt{2}}{2} = \sqrt{2}\) - \(f(x_2) = 2 \sin \left(\frac{\pi}{2}\right) = 2 \times 1 = 2\) - \(f(x_3) = 2 \sin \left(\frac{3\pi}{4}\right) = 2 \times \frac{\sqrt{2}}{2} = \sqrt{2}\) - \(f(x_4) = 2 \sin (\pi) = 2 \times 0 = 0\) - \(f(x_5) = 2 \sin \left(\frac{5\pi}{4}\right) = 2 \times \left(-\frac{\sqrt{2}}{2}\right) = -\sqrt{2}\) - \(f(x_6) = 2 \sin \left(\frac{3\pi}{2}\right) = 2 \times (-1) = -2\) Now, compute the sum: \[ R_6 = \left[\sqrt{2} + 2 + \sqrt{2} + 0 - \sqrt{2} - 2 \right] \times \frac{\pi}{4} \] Simplify inside the brackets: \[ \sqrt{2} + 2 + \sqrt{2} - \sqrt{2} - 2 = (\sqrt{2} + \sqrt{2} - \sqrt{2}) + (2 - 2) = \sqrt{2} + 0 = \sqrt{2} \] So: \[ R_6 = \sqrt{2} \times \frac{\pi}{4} \] Compute numerically: \[ \sqrt{2} \approx 1.414213 \] Thus: \[ R_6 \approx 1.414213 \times \frac{\pi}{4} \] \[ \pi \approx 3.141593 \] \[ R_6 \approx 1.414213 \times \frac{3.141593}{4} \approx 1.414213 \times 0.785398 \] \[ R_6 \approx 1.110720 \] ### **Final answer for (a):** \[ \boxed{ R_6 \approx 1.110720 } \] --- ### **Part (b): Riemann Sum with Midpoints (M6)** --- ### **Step 1: Determine the midpoints of each subinterval** The midpoints are at: \[ x_{k}^{mid} = x_{k-1} + \frac{\Delta x}{2} \] for \(k=1,2,\dots,6\). Since: \[ x_{k-1} = (k-1) \times \frac{\pi}{4} \] then: \[ x_{k}^{mid} = (k-1) \times \frac{\pi}{4} + \frac{\pi}{8} \] which simplifies to: \[ x_{k}^{mid} = \left(k - 1 + \frac{1}{2}\right) \times \frac{\pi}{4} = \left(k - \frac{1}{2}\right) \times \frac{\pi}{4} \] for \(k=1,2,\dots,6\). Calculate each: - \(k=1\): \[ x_{1}^{mid} = \left(1 - \frac{1}{2}\right) \times \frac{\pi}{4} = \frac{1}{2} \times \frac{\pi}{4} = \frac{\pi}{8} \] - \(k=2\): \[ x_{2}^{mid} = \left(2 - \frac{1}{2}\right) \times \frac{\pi}{4} = \frac{3}{2} \times \frac{\pi}{4} = \frac{3\pi}{8} \] - \(k=3\): \[ x_{3}^{mid} = \frac{5\pi}{8} \] - \(k=4\): \[ x_{4}^{mid} = \frac{7\pi}{8} \] - \(k=5\): \[ x_{5}^{mid} = \frac{9\pi}{8} \] - \(k=6\): \[ x_{6}^{mid} = \frac{11\pi}{8} \] --- ### **Step 2: Evaluate \(f(x)\) at each midpoint** Recall: \[ f(x) = 2 \sin x \] Calculate: - \(f\left(\frac{\pi}{8}\right) = 2 \sin \left(\frac{\pi}{8}\right)\) - \(f\left(\frac{3\pi}{8}\right) = 2 \sin \left(\frac{3\pi}{8}\right)\) - \(f\left(\frac{5\pi}{8}\right) = 2 \sin \left(\frac{5\pi}{8}\right)\) - \(f\left(\frac{7\pi}{8}\right) = 2 \sin \left(\frac{7\pi}{8}\right)\) - \(f\left(\frac{9\pi}{8}\right) = 2 \sin \left(\frac{9\pi}{8}\right)\) - \(f\left(\frac{11\pi}{8}\right) = 2 \sin \left(\frac{11\pi}{8}\right)\) Using known sine values: \[ \sin \left(\frac{\pi}{8}\right) \approx 0.382683 \] \[ \sin \left(\frac{3\pi}{8}\right) \approx 0.923880 \] \[ \sin \left(\frac{5\pi}{8}\right) = \sin \left(\pi - \frac{3\pi}{8}\right) = \sin \left(\frac{3\pi}{8}\right) \approx 0.923880 \] \[ \sin \left(\frac{7\pi}{8}\right) = \sin \left(\pi - \frac{\pi}{8}\right) = \sin