I have the joint pdf $f(x_1,x_2,x_3)=12x_2 \;\mathrm f \mathrm o \mathrm r \; 0<x_3<x_2<x_1<1,$ and $0$ elsewhere. I want to find the marginal pdf $f_{x_3}(x_3)$.
I know that to do this I will integrate out $x_1$ and $x_2$, and I have tried setting up the integral as $\int_{x_3}^{1} \int_{x_2}^{1} 12x_2 \;dx_1dx_2$, which yields $4x_3^3 -6x_3^2+2$. This can't be correct since it takes on values greater than one on the interval $(0,1)$.
I can't seem to find where I made a mistake, so if someone could help me understand where I went wrong it would be appreciated!
Your limits are wrong, notice that $x_2$ is bounded from above by $x_1$.