Final answer:
To find the average value of the function p(x) = 4x² + 4x + 2 on the interval 1 ≤ x ≤ 4, calculate the definite integral of the function over the interval and divide it by the length of the interval.
Step-by-step explanation:
To find the average value of the function p(x) = 4x² + 4x + 2 on the interval 1 ≤ x ≤ 4, we need to calculate the definite integral of the function over the given interval and then divide it by the length of the interval.
First, we find the integral of p(x) over the interval [1, 4] using the power rule:
∫(4x² + 4x + 2) dx = [4/3x³ + 2x² + 2x] from 1 to 4
Substituting the upper and lower limits, we get:
[4/3(4)³ + 2(4)² + 2(4)] - [4/3(1)³ + 2(1)² + 2(1)] = 108/3 - 14/3 = 94/3
Finally, we divide this result by the length of the interval (4 - 1 = 3) to get the average value:
Average value = (94/3) / 3 = 94/9 ≈ 10.44