To evaluate the Riemann sum for the function $f(x) = 3 \cos(x)$ over the interval $0 \leq x \leq \frac{3\pi}{4}$ using $n = 6$ subintervals and left endpoints as sample points, we can follow these steps:

Step 1: Find the width of each subinterval.

The total interval is $\left[ 0, \frac{3\pi}{4} \right]$. To find the width of each subinterval, we divide the total length by $n$:

$\Delta x = \frac{\frac{3\pi}{4} - 0}{6} = \frac{3\pi}{24} = \frac{\pi}{8}$

Step 2: Identify the left endpoints of each subinterval.

The left endpoints of the subintervals will be:

$x_0 = 0, \quad x_1 = \frac{\pi}{8}, \quad x_2 = \frac{2\pi}{8} = \frac{\pi}{4}, \quad x_3 = \frac{3\pi}{8}, \quad x_4 = \frac{4\pi}{8} = \frac{\pi}{2}, \quad x_5 = \frac{5\pi}{8}$

Step 3: Evaluate the function $f(x) = 3 \cos(x)$ at each left endpoint.

Now we evaluate $f(x)$ at each left endpoint:

$f(x_0) = 3 \cos(0) = 3 \times 1 = 3$ $f(x_1) = 3 \cos\left(\frac{\pi}{8}\right)$ $f(x_2) = 3 \cos\left(\frac{\pi}{4}\right) = 3 \times \frac{\sqrt{2}}{2} \approx 3 \times 0.70710678118 = 2.121320$ $f(x_3) = 3 \cos\left(\frac{3\pi}{8}\right)$ $f(x_4) = 3 \cos\left(\frac{\pi}{2}\right) = 3 \times 0 = 0$ $f(x_5) = 3 \cos\left(\frac{5\pi}{8}\right)$

Step 4: Set up the Riemann sum.

The Riemann sum is given by the formula:

$S = \sum_{i=0}^{n-1} f(x_i) \Delta x$

Substituting in the values:

$S = \Delta x \left[ f(x_0) + f(x_1) + f(x_2) + f(x_3) + f(x_4) + f(x_5) \right]$

$S = \frac{\pi}{8} \left[ 3 + 3 \cos\left(\frac{\pi}{8}\right) + 2.121320 + 3 \cos\left(\frac{3\pi}{8}\right) + 0 + 3 \cos\left(\frac{5\pi}{8}\right) \right]$

Now, we need to compute the cosine values for the remaining points. I will do that next.

Let me calculate that quickly. math

Given values

pi = math.pi

Function f(x) = 3 * cos(x)

def f(x): return 3 * math.cos(x)

Left endpoints for x

x_values = [0, pi/8, pi/4, 3pi/8, pi/2, 5pi/8]

Calculate f(x) at each left endpoint

f_values = [f(x) for x in x_values]

Riemann sum calculation

delta_x = pi / 8 riemann_sum = delta_x * sum(f_values) riemann_sumThe Riemann sum for the function $f(x) = 3 \cos(x)$ over the interval $0 \leq x \leq \frac{3\pi}{4}$, using $n = 6$ subintervals and left endpoints as sample points, is approximately:

$S \approx 3.099558$

This value is rounded to six decimal places.

Would you like further explanation or have any other questions?

Here are some follow-up questions that could deepen the understanding:

1. How would the Riemann sum change if we used right endpoints instead of left endpoints?
2. What is the difference between a left Riemann sum and a right Riemann sum?
3. How does the number of subintervals ($n$) affect the accuracy of the Riemann sum?
4. What is the exact value of the integral of $f(x) = 3 \cos(x)$ over this interval?
5. How can we approximate the area under the curve using a larger value of $n$?

Tip: Increasing the number of subintervals ($n$) will generally improve the accuracy of the Riemann sum, bringing it closer to the exact integral value.