How to calculate P(X≤x | Y≤y) given joint p.d.f?

Question

I know that $P(X|Y) = \frac{f(x,y)}{f(y)}$ and that $P(X \le x, Y \le y) = \int_{-\infty}^{x} \int_{-\infty}^{y} f(t,s) \space ds \space dt$, but I am not sure how to put them together. The specific p.d.f I am looking at is: $$ f(x,y) = \frac{1}{4}, -1 \le x \le 1 \quad \text{and} \quad x^4 \le y^2 \le (x^2+1)^2 $$

Answer 1

This is just an extended comment. To plan for the limits of integration plotting the joint and marginal densities is helpful.

Here is the joint density (which has a value of 1/4 over the shaded region and zero elsewhere):

And the marginal density of $Y$: