How can we plot an equation in 3 variables on a 2D plane in context with functions of pair of Random Variables?

Question

I came across the following question:

The Question

I tried solving it, the following is my attempt:

$$ P[W\le w] = P[XY\le w] = P[Y\le w/X] $$

And then I simply double integrated keeping the limits of X on the outer integral and between 0 and 1, and that of Y between 0 and $w/x$. But the answer is wrong 'cause finally I ended up calculating $$[w*ln(x)]$$ between 0 and 1.

The solution as given in Probability and Stochastic Processes-Roy D. Yates

1. Why is my answer wrong?
2. What I'm not able to understand is that the plot given should be a 3D plot, how can it be represented in a 2D plane?

Answer 1

(1) $\mathsf{P}(Y\le w/X)=\int_0^1\int_0^{1\wedge w/x}1\,dy\,dx=w(1-\ln w)$ for $w\in [0,1]$.

(2) It is a level curve of the function $f(x,y)=xy$.