First Price Auction (Expected Payoff) I'm trying to solve followed first-price auction problem.

Bidder's pdf is
$$
f(v_i)= \begin{cases}
\dfrac{1}{8}v_i, &amp; \text{if} &amp; 0\leq v_i\leq4\\
0, &amp; \text{if} &amp;  \text{otherwise}\\
\end{cases}
$$

Bidders only know their own values. One gets 0 if lose the auction and ($v_i-b_i$) if win the auction game.

The question asks the symmetric bidders' bid under Bayesian Nash equilibrium and expected payoff of bidders and seller.

So, one bidder will maximize as follows:

$$\max_{b_1}\,\,(v_1-b_1)\Pr(b_1&gt;b_2)$$

If we take the derivative with respect to $b_1$ we get

$$b(v_1)=\dfrac{\int v_1f(v_1)\,dv_1}{F(v_1)}$$

We know from question $f(v_1)=\dfrac{1}{8}v_i$ and we can find $F(v_1)=\dfrac{1}{16}v_i^2$. If we replace them, we get

$$b(v_1)=\dfrac{\int v_1\dfrac{1}{8}v_1\,dv_1}{\dfrac{1}{16}v_1^2} = \dfrac{2}{3}v_1$$

So this is the bidding of bidder 1. However, I couldn't find the expected payoff bidder 1 and bidder 2. My question is how can I use PDF or CDF to find the expected payoffs of the bidders.
Let me answer here putting together all the hints in the comments.

You've figured out that the equilibrium bid for a type $v$ bidder is $b(v) = \frac{2}{3}v$.

The bids are strictly increasing in $v$, so bidder $i$ wins whenever $b(v_i) \geq b(v_j)$ or $v_i \geq v_j$.

The expected payoff for a type $v$ bidder is thus
$$
(v -b(v))P(b_1 \geq b_2)
$$ 
or
$$
(v - \frac{2}{3}v)P(v \geq v_2) = \frac{1}{3}P(v \geq v_2)
$$
So what is the probability that $v \geq v_2$?

By definition of CDF $P(x \leq v) =  F(v) = \int_0^v f(x) dx$

So $P(v_2 \leq v) = F(v)$.

Thus expected utility for a type $v$ bidder is:
$$
u(v) = \frac{v}{3} F(v)
$$