Calculating the Average Value of a Function Between Two Limits

[Justabeginner]

Homework Statement

What is the average value of a function 1/x between x=2/3 and x=8/3?

Homework Equations

1/(b-a) ∫f(x) dx with a and b being the lower and upper limits, respectively

The Attempt at a Solution

1/([8/3] - [2/3])∫1/x dx
1/(6/3) ∫1/x dx
1/2 ∫1/x dx
1/2 * (ln x)

Plug in:
(ln {8/3})/2 - (ln {2/3}/2)
ln 4/2
ln 2

Is this correct? Thanks :)

 

[Mandelbroth]

Homework Statement

What is the average value of a function 1/x between x=2/3 and x=8/3?

Homework Equations

1/(b-a) ∫f(x) dx with a and b being the lower and upper limits, respectively

The Attempt at a Solution

1/([8/3] - [2/3])∫1/x dx
1/(6/3) ∫1/x dx
1/2 ∫1/x dx
1/2 * (ln x)

Plug in:
(ln {8/3})/2 - (ln {2/3}/2)
ln 4/2
ln 2

Is this correct? Thanks :)

Looks good.

Though, just to be clear, your reasoning for your last few steps was $\frac{1}{2}\ln\left(\frac{(\frac{8}{3})}{(\frac{2}{3})}\right) = \frac{1}{2}\ln(4) = \ln(4^{\frac{1}{2}}) = \ln2$, correct?

 

[Justabeginner]

Yes, that is exactly what I have written on my paper here. Thank you so much for your help :)