Evaluate Integral using concept of Anti-derivatives

[David S.]

Can someone please help me with this Calculus Question?

 

[Romsek]

two things you need to know

i) over an interval [MATH][a,b][/MATH] the average of [MATH]f(x)[/MATH] over that interval is given by
[MATH]\bar{f}_{[a,b]} =\dfrac{1}{b-a}\displaystyle \int \limits_a^b f(x)~dx[/MATH]
ii) for a function [MATH]f(x)[/MATH] with a anti-derivative [MATH]F(x), \displaystyle \int \limits_a^b f(x)~dx = F(b) - F(a)[/MATH]

 

[David S.]

So would the answer be this? I'm not sure how ii) fits into the question.

 

[Romsek]

[MATH]F(x) = e^{2x}\\ \bar{f} = \dfrac{1}{4} \left(e^{4}-e^{-4}\right) = \dfrac 1 2 \cosh(4)[/MATH]
Your answer is correct. Just written in a bizarre form.

 

[David S.]

Ok thanks