This is Problem 6.10 in Mathematical Statistics with Applications, 5th Ed., by Wackerly, Mendenhall, and Scheaffer.
Problem Statement: In a process of sintering (heating) two types of copper powder the density function for $Y_1,$ the volume proportion of solid copper in a sample, is given by $$f_1(y_1)= \begin{cases} 6y_1(1-y_1),&0\le y_1\le 1\\ 0,&\text{elsewhere.} \end{cases} $$ The density function for $Y_2,$ the proportion of type $A$ crystals among the solid copper, is given as $$f_2(y_2)= \begin{cases} 3y_2^2,&0\le y_2\le 1\\ 0,&\text{elsewhere.} \end{cases} $$ The variable $U=Y_1Y_2$ gives the proportion of the sample volume due to type $A$ crystals. If $Y_1$ and $Y_2$ are independent, find the probability density function for $U.$
My Work So Far: Since the variables are independent by assumption, the joint density function is $$f(y_1,y_2)= \begin{cases} 18y_1y_2^2(1-y_1),&0\le y_1,y_2\le 1\\ 0,&\text{elsewhere.} \end{cases} $$ Now then, we have \begin{align*} F_U(u) &=P(U\le u)\\ &=P(Y_1Y_2\le u)\\ &=P(Y_1\le u/Y_2)\\ &=\int_0^1\int_0^{u/y_2}18y_1y_2^2(1-y_1)\,dy_1\,dy_2. \end{align*}
My Question: The problem is that the integral in my last line does not converge, no-how, no-way. I've tried reversing the order of integration, to no avail. Am I missing something basic?
Thanks for your time!