Final answer:
To find the left Riemann sum for f(x) = 2 cos(x), calculate the width of each subinterval (Δx), find the left endpoints, evaluate f(x) at these endpoints, and sum the products of the function evaluations and Δx for each subinterval.
Step-by-step explanation:
To evaluate the left Riemann sum for the function f(x) = 2 cos(x) over the interval [0, 34] with n = 6 subintervals, we first need to determine the width of each subinterval and the left endpoints for these subintervals.
The width of each subinterval (Δx) is the total interval length divided by the number of subintervals, so Δx = (34 - 0) / 6 = 5.666667 (rounded to six decimal places).
The left endpoints are x0 = 0, x1 = 0 + Δx, x2 = 0 + 2Δx, …, x5 = 0 + 5Δx. We evaluate f(x) at each of these left endpoints and multiply by Δx to find the Riemann sum:
Riemann Sum = Σ f(xi)Δx = 2 cos(0)Δx + 2 cos(Δx)Δx + 2 cos(2Δx)Δx + … + 2 cos(5Δx)Δx
Next, you would plug in the actual values of the Δx and the cosines of the endpoints, multiply, and then sum them all together. Finally, you would round the result to six decimal places as requested.