Question

Let f(x) = x3 and compute the Riemann sum of f over the interval [1,2], using the following number of subintervals (n). In each case, choose the representative points to be the left endpoints of the subintervals. (Round your answers to two decimal places.)

(a) Use two subintervals of equal length (n = 2).

Answer 1

To compute the Riemann sum of the function f(x) = x^3 over [1,2] with two subintervals, we evaluate the function at the left endpoints of subintervals with width 0.5, giving a Riemann sum of 2.19 after rounding to two decimal places.

Step-by-step explanation:

To calculate the Riemann sum for the function f(x) = x3 over the interval [1,2] with two subintervals (n = 2), first, we need to determine the width of each subinterval, which is the length of the interval [1,2] divided by the number of subintervals, that is, (2 - 1) / 2 = 0.5. The left endpoints for these subintervals are x1 = 1 and x2 = 1.5. We then evaluate the function at these left endpoints and multiply by the width of the subintervals to get the approximate area under the curve.

The Riemann sum S is calculated as: S = f(x1)*width + f(x2)*width = (1)3*0.5 + (1.5)3*0.5 = 0.5 + 1.6875 = 2.1875. Rounded to two decimal places, the Riemann sum is 2.19.