Question

If the average value of the function f on the interval 2 ≤ x ≤ 6 is 3, what is the value of the integral [2,6] (5f(x) + 2)dx?

a) 16
b) 22
c) 34
d) 44

Answer 1

The value of the integral [2,6] (5f(x) + 2)dx is 12.

Step-by-step explanation:

To find the value of the integral, we can use the average value theorem for integrals. The average value theorem states that if the average value of a function f on an interval [a, b] is A, then the definite integral of f from a to b is equal to A times the length of the interval (b - a).

In this case, the average value of the function f on the interval 2 ≤ x ≤ 6 is 3. So, the definite integral of (5f(x) + 2) from 2 to 6 is equal to 3 times the length of the interval (6 - 2).

Therefore, the value of the integral [2,6] (5f(x) + 2)dx is 4 ∙ 3 = 12.