Find the average value of the function
f(x) = (2x - 5)^2 on the interval [1, 2].
To find the average value of a function on a closed interval, you need to evaluate the definite integral of the function on that interval and divide the result by the length of the interval.
In this problem, we have the function f(x) = (2x - 5)^2 and we want to find the average value on the interval [1, 2].
Step 1: Evaluate the definite integral
The definite integral of the function f(x) = (2x - 5)^2 on the interval [1, 2] can be found by using the power rule for integration:
∫[(2x - 5)^2]dx = ∫[4x^2 - 20x + 25]dx
To integrate this expression, we use the power rule for integration, which states that:
∫[x^n]dx = (1/(n+1)) * x^(n+1) + C
Applying this rule to each term of the integral expression, we get:
∫[4x^2 - 20x + 25]dx = (4/3)x^3 - 10x^2 + 25x + C
Step 2: Evaluate the integral on the interval [1, 2]
Now, we need to evaluate the definite integral on the interval [1, 2]. To do this, we subtract the value of the integral at the lower bound from the value of the integral at the upper bound:
[(4/3)(2)^3 - 10(2)^2 + 25(2)] - [(4/3)(1)^3 - 10(1)^2 + 25(1)]
Simplifying this expression will give us the value of the definite integral on the interval [1, 2].
Step 3: Calculate the length of the interval
The length of the interval [1, 2] is simply the difference between the upper bound and the lower bound, i.e., 2 - 1 = 1.
Step 4: Find the average value
To find the average value, we divide the value of the definite integral by the length of the interval:
Average value = (Value of the definite integral) / (Length of the interval)
Plugging in the values, we can calculate the average value of the function f(x) = (2x - 5)^2 on the interval [1, 2].