Consider an example of a risk-neutral seller that has two distinct indivisible goods for sale. The seller wants to maximize the expected revenue. The buyer's utility is $$I_av_a+I_bv_b-t,$$ where $I_a,I_b$ are indicators of whether the buyer gets good $a$ or $b$ and $t$ is the monetary transfer. Both $v_a,v_b$ are the types of the buyer (willingness to buy the good). Let $v_a$ and $v_b$ be i.i.d. draws from a uniform distribution on $[0,1]$.
Assume a seller quotes three prices: $p_a,p_b,p_{ab}$, where $p_{ab}\leq p_a+p_b$. What is the optimal choice of $p_a,p_b,p_{ab}$?
The textbook (T. Borgers, An Intro to the theory of Mechanism Design) I am studying simply says that this is a simple calculus problem and gives the solution as $$p_a=p_b=2/3\\ p_{ab}=\frac{4-\sqrt{2}}{3}.$$
I am not sure how to do that. I'd appreciate any help or hint. Thank you.