The measured radius of a circle has the pdf $f(r)=6r(1-r)$ for $0<r<1$. Find the expected circumference of the circle.
I am not sure what to do here, but this is what I tried:
Since $c=2\pi r$, I just substituted $2\pi r$ and got the function $12\pi(r-2r^{2})$, but I am not sure whether this is the correct approach.
You should compute the following quantity.
$$E[C] = \int_0^1 (2\pi r) (6r)(1-r) \, dr = 12\pi \int_0^1 r^2-r^3 \, dr$$