Question

Use the Midpoint Rule and the given data to estimate the value of the integral I. (Give the answer to two decimal places.)

Answer 1

Answer:

a) integral = 24.72

b) |Error| ≤ 0.4267

Step-by-step explanation:

a)

The integral:

$\int_{0}^{3.2} f(x) dx$

can be  approximated with the midpoint rule, as follows:

6.7*(0.8 - 0.0) + 8.9*(1.6 - 0.8) + 6.9*(2.4-1.6) + 8.4*(3.2 - 2.4) = 24.72

(that is, all the intervals are 0.8 units length and f(x) is evaluated in the midpoint of the interval)

b)The error bound for the midpoint rule with n points is:

|Error| ≤ K*(b - a)^3/(24*n^2)

where b and are the limits of integration of the integral and K = max |f''(x)|

Given that -5 ≤ f''(x) ≤ 1, then K = 5.  Replacing into the equation:

|Error| ≤ 5*(3.2 - 0)^3/(24*4^2) = 0.4267