Question

Let f(x) = x2, and compute the Riemann sum of f over the interval [3, 5], choosing the representative points to be the midpoints of the subintervals and using the following number of subintervals (n). (Round your answers to two decimal places.) N=2

Answer 1

To compute the Riemann sum of the function f(x) = x^2 over the interval [3, 5] with midpoints as representative points and n = 2 subintervals, divide the interval into subintervals, calculate the width of each subinterval, find the representative points, evaluate the function at each point, and multiply by the width of the subinterval. The Riemann sum is 32.50.

Step-by-step explanation:

To compute the Riemann sum of the function f(x) = x^2 over the interval [3, 5] with a choice of representative points as the midpoints of the subintervals, and using n = 2 subintervals, follow these steps:

1. Divide the interval [3, 5] into n subintervals of equal length.
2. Calculate the width of each subinterval: Δx = (b - a) / n = (5 - 3) / 2 = 1.
3. Identify the representative points for each subinterval. In this case, since we are using midpoints, the representative points for the two subintervals are 3.5 and 4.5.
4. Evaluate the function at each representative point. f(3.5) = (3.5)^2 = 12.25 and f(4.5) = (4.5)^2 = 20.25.
5. Multiply each function value by the width of the corresponding subinterval. (12.25 * 1) + (20.25 * 1) = 32.50.

Therefore, the Riemann sum of f over the interval [3, 5] with n = 2 subintervals, using midpoints as representative points, is 32.50.