density of the product of uniformly distributed random variables Let $X$ and $Y$ be  independent with uniform distribution over $(0,a)$ and set $Z=X^2Y^2$. What is the joint density of $Z$
\begin{align}
F_{Z}(t) &= \mathbb P(X^2Y^2<t)
= \begin{cases}
0,& t<0\\
1,& t>a^4
\end{cases}
\end{align}
Consider the case when $0<t<a^4$
$$F_{Z}(t) = \mathbb P(XY<\sqrt t)=\int\int_{xy<\sqrt t}f(x,y)dxdy=\int\int_{xy<\sqrt t}f_X(x)f_{Y}(y)dxdy =1/a^2 \int_{\sqrt t/a}^{a}dx\int_{\sqrt t/x}^{a}dy$$
Are the integration limits set correctly?

No. You should integrate below the hyperbola $yx=\sqrt{t}$ or $y=\frac{\sqrt{t}}{a}$ at the picture:

So 
$$
F_{Z}(t) = \mathbb P(XY<\sqrt t)=1/a^2 \int_0^{\sqrt t/a}dx\int_0^a dy + 1/a^2  \int_{\sqrt t/a}^a dx\int_0^{\sqrt t/x} dy 
$$
Here is slightly modified picture with integration bounds for $y$ are drawn at each $x$. 

