Find the value for k that gives a level of significance of 0.05, when given a null hypothesis that is to be rejected when $y_1*y_2 \leq k$

Question

A random sample of size 2 is drawn from a uniform pdf defined over the interval $[0, θ ]$. We wish to test $H_0 : θ = 2 $ versus $H_1 : θ < 2$ by rejecting $H_0$ when $y_1y_2≤k^∗$. Find the value for k that gives a level of significance of 0.05.

I know that the joint pdf $f_{{Y_1},{Y_2}}(y_1,y_2)=1/4$ where $0\leq y_1 \leq2$ and $0\leq y_2 \leq2$. I also know that the level of significance which is also the type I error is $P(y_1*y_2\leq k|\theta=2)$. How do I proceed finding $P(y_1*y_2\leq k|\theta=2)$.

Answer 1

Sketch the region of integration bounded by the curves

$$y_1=0, y_2=0, y_1=2, y_2=2, y_1y_2=k$$

to get

$$0.05 = \int_0^{\frac k 2} \int_0^{2} \frac14 dy_2 dy_1 + \int_{\frac k 2}^{2} \int_0^{\frac k {y_1}} \frac14 dy_2 dy_1$$

$$ = \int_0^{2} \int_0^{\frac k 2} \frac14 dy_1 dy_2 + \int_{0}^{2} \int_0^{\frac k {y_2}} \frac14 dy_1 dy_2$$