How do you find the average value of the function for #f(x)=e^x, -1<=x<=1#?

1 Answer
Steve M
Apr 28, 2018

# bar (f)(x) = 1/2 \ (e-1/e) ~~ 1.1752 #

Explanation:

By definition, the average value of a function #f(x)# over a domain #[a,b]# is given by:

# bar (f)(x) = 1/(b-a) \ int_a^b \ f(x) \ dx #

So, for the given function, #f(x)=e^x# with #-1 le x le 1#, we have:

# bar (f)(x) = 1/(1-(-1)) \ int_(-1)^(1) \ e^x \ dx #

# \ \ \ \ \ \ \ = 1/2 \ [e^x]_(-1)^(1) #

# \ \ \ \ \ \ \ = 1/2 \ (e-e^(-1)) #

# \ \ \ \ \ \ \ = 1/2 \ (e-1/e) #

# \ \ \ \ \ \ \ ~~ 1.1752 #

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