Let $U_1$, $U_2$, and $U_3$ be identical, independent, random variables distributed $Uniform(0, 1)$. Let $L$ be the minimum and $M$ be the maximum of $U_1$, $U_2$, and $U_3$.
I want to find the marginal probability density function (PDF) of $L$ ($f_L(l)$). I got that the joint PDF of $L$ and $M$ is $f_{L, M}(l, m) = 6(m - l)$. I'm pretty sure this is correct.
Next, I tried integrating out the $m$ to get $f_L(l)$:
$ f_L(l) = \int_0^1 6m - 6l~dm \\ = 3m^2 - 6lm \text{ at m = 1 and m = 0.} \\ = 3 - 6l $
However, I am pretty sure that $f_L(l) = 3(1-l)^2$ just reasoning out what the CDF of $L$ should be $(1 - l)^3$.
Why is my integration wrong?