Final answer:
The average value of the function f(x) = x² - 1/x over the interval [1, 9] is found by integrating the function over the interval and then dividing by the interval's length.
Step-by-step explanation:
To find the average value of the function f(x) = x² - 1/x over the interval [1, 9], we must integrate the function over the interval and then divide by the length of the interval. The average value of a function f(x) from a to b is given by the formula:
Average value = (1 / (b - a)) ∫ f(x) dx
Here, a = 1 and b = 9. We'll perform the following steps to compute the average value:
- First, calculate the definite integral of f(x) from 1 to 9.
- Then, divide the result by 9 - 1, which is the length of the interval.
By performing the integration, we get the following result:
Integral of f(x) from 1 to 9 = ∫ (x² - 1/x) dx
This integration gives us a value, which we then divide by 8 to get the average value of the function over the given interval.