Find the average value of the function over the given interval. (Round your answer to four decimal places.) f(x) = 4 − x2, [−2, 2]

Question

Find all values of x in the interval for which the function equals its average value. (Enter your answers as a comma-separated list. Round your answers to four decimal places.)

(the title is part a to the question and the description is part b)

Answer 1

a) To find the average value of a function on [a, b], find the definite integral of the function on the interval with lower bound a and upper bound b and divide it by the length of the interval.

In this case, the interval is [-2, 2], which has length 4.

Use basic antidifferentiation techniques to find the antiderivative of 4 - x^2 and apply the FTC.

Don't forget to divide by 4.

b) Once you have the average value, let's call it K, solve the equation f(x) = K. Since f(x) is a parabola, you should expect two solutions.

Answer 2

The average is (1/4) of the integral from -2 to 2 of f(x) dx.

Then take that value and set it equal to f(x) and solve for x.

If you need additional help, please comment.