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] 

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.) 

X = _______ 

Answer 1

Answers are

Average value = 32/3 = 10.66666666 = 10.6667 ( Rounded to four decimals)

x = , = -2.30940108 , 2.30940108 = -2.3094 , 2.3094 (Rounded to four decimals )

Answer 2

SOLUTION :

Average value (f average)  of the given function f(x) in interval (-4, 4)

    4

=  ∫    f(x) dx  / (4 - (-4))

 - 4

            4

= 1/8   ∫   (16 - x^2)  dx

          - 4

                                    4

= 1/8 *  [ 16x - x^3 /3] ]                            

                                  - 4                                                     

= 1/8 * [ (16*4 - 4^3/3) - (16*(-4) - 4(-4)^3 / 3))

= 1/8 [128/3 + 128/3]

=   32/3 = 10.6667 

Value of x for which f(x) = f average = 32/3 :

f(x) = 16 - x^2 = 32/3

=> x^2 - 16 + 32/3 = 0

=> x^2 - 16/3 = 0

=> x = +/- sqrt(16/3) = +/- 2.3094 

=> x = - 2.3094, 2.3094 (ANSWER).