Answer:
a) integral = 24.72
b) |Error| ≤ 0.4267
Step-by-step explanation:
a)
The integral:
[tex]\int_{0}^{3.2} f(x) dx[/tex]
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
In this exercise we have to use our knowledge of equations to calculate the values of the integral and its error, so we have to:
a) [tex]f(x)= 24.72[/tex]
b) [tex]|Error| \leq 0.4267[/tex]
a)So first we must use the integral to find the values of X, then we will use the midpoint rule, as follows:
[tex]f(x)=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[/tex]
b)The error bound for the midpoint rule with n points is:
[tex]|Error| \leq K*(b - a)^3/(24*n^2)\\|Error| \leq 5*(3.2 - 0)^3/(24*4^2) = 0.4267[/tex]
See more about equations at brainly.com/question/2263981
