Final answer:
To calculate the average value of the function p(x) = 6x² + 2x + 2 from x = 1 to x = 6, integrate p(x) over this interval and then divide by the interval length, which is 5. After solving the integral and evaluating the limits, divide the result by 5 to get the average value.
Step-by-step explanation:
To find the average value of the function p(x) = 6x² + 2x + 2 on the interval 1 ≤ x ≤ 6, you use the formula for the average value of a function, which is given by:
Average value = ±b±a f(x) dx / (b - a)
For the function p(x), the limits of integration a and b are 1 and 6, respectively. After integrating the function, you will divide by the length of the interval, which is b - a = 6 - 1 = 5. The integration yields:
±1±6 (6x² + 2x + 2) dx = [(2x³ + x² + 2x)]61
After evaluating this at x = 6 and x = 1, we subtract the results and then divide by 5. The final number is the average value of p(x) over the given interval.