Question

Find the average value of the function over the given interval. (Round your answer to four decimal places.)

f(x) = 16 − x², [−4, 4]

Answer 1

To find the average value of a function over an interval, integrate the function over that interval and divide the result by the length of the interval.

Step-by-step explanation:

To find the average value of a function over an interval, you need to integrate the function over that interval and divide the result by the length of the interval.

In this case, the function is f(x) = 16 - x² and the interval is [-4, 4].

First, find the integral of f(x) by using the power rule of integration: ∫(16 - x²) dx = 16x - (x³/3).

Next, evaluate the definite integral over the interval [-4, 4] by substituting the upper and lower limits into the integral: ∫[-4,4](16 - x²) dx = [16x - (x³/3)]-4⁴. The result is the average value of the function over the given interval.