Let f(x) = x2, and compute the Riemann sum of f over the interval [8, 10]

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

a) n=2 (8.5)^2 +(9.5)^2 = 72.25 +90.25 = 162.5

d) compare to the integral of x^2 = x^3/3

evaluated from 8 to 10 = 1000/3 - 512/3= 488/3= 162.666...

error of only 0.1666...

b)n=5 midpoints are 8.2, 8.6, 9, 9.4, 9.8

area = .4(8.2^2 +8.6^2 + 9^2 + 9.4^2 + 9.8^2)= 162.64

which is even closer, only off by 0.02666...

c) n=10, midpoints are 8.1, 8.3, 8.5, 8.7, 8.9, 9.1, 9.3, 9.5, 9.7, 9.9

square each, sum them, then multiply by .2

=162.66

within 0.00666... of the exact area