Final answer:
The value of a in the function f(x) = ax² - 2ax + a, set the average value of this function over the interval [1, 4] to 13. Integrate the function and solve the equation to determine the value of a, which is approximately 3.59.
Step-by-step explanation:
To find the value of a in the function f(x) = ax² - 2ax + a, we need to determine the average value of this function over the interval [1, 4] and set it to 13. The average value of a function over an interval is given by the integral of the function over the interval divided by the length of the interval.
So, we have the equation:
13 = (1/3) * ∫(ax² - 2ax + a) dx [from 1 to 4]
By integrating and solving the equation, we can find the value of a. Let's proceed with the integration:
13 = (1/3) * [⅓ax³ - ax² + ax] [from 1 to 4]
This simplifies to:
13 = a(4³/3 - 4²/3 + 4/3) - a(1³/3 - 1² + 1/3)
After evaluating the integrals and solving the equation, we find that a = 3.59.