Expert: Paul Klarreich - 2/23/2009

Question
Find the average value of the function over the given interval and all values of x in the interval for which the function equals its average value.

1) f(x)=4-x^2, [-2,2]

When I did it I got average value of 8/3 and
x=+or- 2√(3)/3 and 1.155 but i am not sure if that is correct so if you could show me your answer and how you got it that would be great...Thanks

Answer
Questioner:   Melanie
Category:  Calculus
Private:  No
 
Subject:  Integration
Question:  Find the average value of the function over the given interval and all values of x in the interval for which the function equals its average value.

1) f(x)=4-x^2, [-2,2]

When I did it I got average value of 8/3 and
x=+or- 2√(3)/3 and 1.155 but i am not sure if that is correct so if you could show me your answer and how you got it that would be great...Thanks
.......................
Hi, Melanie,

Looks good from here.  

Average value of f(x) over [a,b] = INTEGRAL of f(x)/(b-a)

{2
|  (4 - x^2)dx =
}-2

(4x - x^3/3) [-2 to 2] =

8 - 8/3 - [-8 + 8/3] = 16 - 16/3 = 32/3, divided by 4 = 8/3.

4 - x^2 =  8/3

x^2 = 4/3  x = +- 2/sqrt(3)