Question

Let f(x) = x2, and compute the Riemann sum of f over the interval [8, 10], 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.) (a) two subintervals of equal length (n = 2) (b) five subintervals of equal length (n = 5) (c) ten subintervals of equal length (n = 10) (d) Can you guess at the area of the region under the graph of f on the interval [8, 10]

Answer 1

To compute the Riemann sums for f(x) = x^2 over [8, 10], one can divide the interval into subintervals, find the midpoints, and calculate areas of rectangles. The sums for n=2, n=5, and n=10 are roughly 162.50, 163.68, and 163.98 respectively. As n increases, the sums approximate the definite integral of f(x) which gives the actual area under the curve.

Step-by-step explanation:

Computing the Riemann sum of f(x) = x2 over the interval [8, 10] involves finding the total sum of areas of rectangles under the curve with the height determined by the value of the function at the midpoint of each subinterval.

1. n = 2: With two subintervals, each has a length of 1 ((10 - 8) / 2). The midpoints are 8.5 and 9.5, so we calculate the areas as (8.52)*1 + (9.52)*1 = 72.25 + 90.25, yielding a Riemann sum of 162.50.
2. n = 5: With five subintervals, each has a length of 0.4 ((10 - 8) / 5). The midpoints are 8.2, 8.6, 9.0, 9.4, and 9.8. Thus, we sum (8.22)*0.4 + (8.62)*0.4 + (9.02)*0.4 + (9.42)*0.4 + (9.82)*0.4 to get a Riemann sum of 163.68.
3. n = 10: With ten subintervals, each has a length of 0.2 ((10 - 8) / 10). Adding up the areas for midpoints 8.1, 8.3, ..., 9.9 gives us a Riemann sum of 163.98.
4. Estimating the area under the graph of f(x) from x = 8 to x = 10, as n increases, the Riemann sum approaches the actual area which can be computed using the definite integral of f(x) from 8 to 10 which is (103/3) - (83/3) = 332.67, rounded to two decimal places.