To compute the Riemann sum $R_6$ for the function $f(x) = 5 \sin(x)$ over the interval $[0, \frac{3\pi}{2}]$ with 6 subintervals, using right endpoints, follow these steps:

Step 1: Determine the width of each subinterval.

The width of each subinterval, denoted by $\Delta x$, is calculated as:

$\Delta x = \frac{b - a}{n} = \frac{\frac{3\pi}{2} - 0}{6} = \frac{\frac{3\pi}{2}}{6} = \frac{\pi}{4}$

Step 2: Identify the sample points (right endpoints).

The right endpoints are given by:

$x_i = a + i\Delta x$

For $i = 1, 2, \dots, 6$, the right endpoints are:

$x_1 = \frac{\pi}{4}, \quad x_2 = \frac{\pi}{2}, \quad x_3 = \frac{3\pi}{4}, \quad x_4 = \pi, \quad x_5 = \frac{5\pi}{4}, \quad x_6 = \frac{3\pi}{2}$

Step 3: Calculate the function values at these points.

We now evaluate $f(x) = 5 \sin(x)$ at each of these right endpoints:

$f(x_1) = 5 \sin\left(\frac{\pi}{4}\right), \quad f(x_2) = 5 \sin\left(\frac{\pi}{2}\right), \quad f(x_3) = 5 \sin\left(\frac{3\pi}{4}\right)$ $f(x_4) = 5 \sin(\pi), \quad f(x_5) = 5 \sin\left(\frac{5\pi}{4}\right), \quad f(x_6) = 5 \sin\left(\frac{3\pi}{2}\right)$

Using the values of $\sin(x)$:

$f(x_1) = 5 \times \frac{\sqrt{2}}{2}, \quad f(x_2) = 5 \times 1, \quad f(x_3) = 5 \times \frac{\sqrt{2}}{2}$ $f(x_4) = 5 \times 0, \quad f(x_5) = 5 \times \left(-\frac{\sqrt{2}}{2}\right), \quad f(x_6) = 5 \times (-1)$

Simplifying these:

$f(x_1) = 5 \times 0.707106 = 3.535534, \quad f(x_2) = 5, \quad f(x_3) = 3.535534$ $f(x_4) = 0, \quad f(x_5) = -3.535534, \quad f(x_6) = -5$

Step 4: Compute the Riemann sum.

The Riemann sum is given by:

$R_6 = \Delta x \left[ f(x_1) + f(x_2) + f(x_3) + f(x_4) + f(x_5) + f(x_6) \right]$

Substitute the values of $f(x_i)$ and $\Delta x = \frac{\pi}{4}$:

$R_6 = \frac{\pi}{4} \left[ 3.535534 + 5 + 3.535534 + 0 - 3.535534 - 5 \right]$ $R_6 = \frac{\pi}{4} \left[ 3.535534 + 5 + 3.535534 - 3.535534 - 5 \right]$ $R_6 = \frac{\pi}{4} \times 3.535534 = \frac{\pi}{4} \times 3.535534$ Now, calculate the result:

$R_6 \approx 2.774$

Thus, the Riemann sum $R_6$ is approximately $2.774$.

Would you like more details on any of the steps? Here are some related questions:

1. What is the general formula for the Riemann sum?
2. How do you find left and midpoint Riemann sums?
3. Can the number of subintervals affect the accuracy of Riemann sums?
4. What happens when using non-uniform subintervals?
5. How is the Riemann sum related to the definite integral?

Tip: Increasing the number of subintervals generally increases the accuracy of the Riemann sum approximation to the integral.