Joint Uniform Distrubtion of 3 variables problem

Hi all,

I am working through a problem but I cant seem to see the method... the answer is in the back of the text book but no method!

Q: Let $\displaystyle U_1, U_2, U_3$ be uniformly distributed on $\displaystyle (0,1)$ . Find the prob that $\displaystyle U_1x^2 + U_2x +U_3$ have real roots. The answer in the back of the book is 1/9.

So using the quadratic eqn, I deduce taht the prob distribution we need to find is given by:

$\displaystyle P(U_2^2 -4U_2U_3 >0)$...... I am not sure how to evaluate this at all.

Looking through the usual methods in the text book usually points to evaluating the area via direct integration but when I integrate:

$\displaystyle \int_0^1 \int_0^1 \int_0^1 U_2^2 -4U_2U_3 du_1 du_2 du_3 $ I get a negative answer which is obviously wrong... I suspect the limits need to be changed, but not sure where to start.....

Thanks for reading (Nod)