f(x) = 3 cos(x)    0 ≤ x ≤ 3𝜋/4evaluate the Riemann sum with n = 6, taking the sample points to be left endpoints. (Round your answer to six decimal places.)

To evaluate the Riemann sum with n = 6, we first need to determine the width of each subinterval. The interval is from 0 to 3𝜋/4, so the width (Δx) is (3𝜋/4 - 0)/6 = 𝜋/8.

Next, we need to determine the left endpoints of each subinterval. Since we're starting at 0 and each subinterval has a width of 𝜋/8, the left endpoints are 0, 𝜋/8, 𝜋/4, 3𝜋/8, 𝜋/2, and 5𝜋/8.

Now we can evaluate the function f(x) = 3 cos(x) at each of these points:

f(0) = 3 cos(0) = 3 f(𝜋/8) = 3 cos(𝜋/8) ≈ 2.61312593 f(𝜋/4) = 3 cos(𝜋/4) ≈ 2.12132034 f(3𝜋/8) = 3 cos(3𝜋/8) ≈ 1.47814376 f(𝜋/2) = 3 cos(𝜋/2) = 0 f(5𝜋/8) = 3 cos(5𝜋/8) ≈ -0.47814376

Finally, we can calculate the Riemann sum by multiplying each function value by the width of the subintervals and adding up the results:

Riemann sum = Δx [f(0) + f(𝜋/8) + f(𝜋/4) + f(3𝜋/8) + f(𝜋/2) + f(5𝜋/8)] = 𝜋/8 [3 + 2.61312593 + 2.12132034 + 1.47814376 + 0 - 0.47814376] = 𝜋/8 [8.73444627] ≈ 3.433611

So, the Riemann sum of f(x) = 3 cos(x) from 0 to 3𝜋/4 with n = 6 and left endpoints is approximately 3.433611.