Question

Evaluate the left Riemann sum of the function f(x) = cos(x) on the interval [0, 34] with n = 6, taking the sample points to be the left endpoints. Round your answer to six decimal places.

Answer 1

The left Riemann sum for f(x) = cos(x) over [0, π/4] with n = 6 is calculated by dividing the interval into equal parts, identifying left endpoints, evaluating the function at those points, then summing the products of these values with the subinterval width.

Step-by-step explanation:

To calculate the left Riemann sum for the function f(x) = cos(x) over the interval [0, π/4] with n = 6 subdivisions, we need to follow these steps:

1. Divide the interval into 6 equal subintervals.
2. Determine the width (Δx) of each subinterval by subtracting the lower limit of the integral from the upper limit, and then dividing by the number of subintervals.
   Δx = ∆(π/4 - 0)/6
3. Identify the left endpoint of each subinterval, which will be our sample points for the Riemann sum.
4. Evaluate the function f(x) at each left endpoint.
5. Sum up all the values obtained in step 4, each multiplied by Δx.

Since the function cos(x) is well-behaved on the interval [0, π/4], we can expect a good approximation of the integral by calculating the left Riemann sum.