Second Price Auction - adjusting PDF for reservation price
To find expected revenue of the seller in a second price auction with reserved price consisting of two bidders who bid their valuations in equilibrium, we do the following :

Given that valuations are i.i.d with pdf $f$ and CDF $F$ over $[0, \hat{v}]$, the revenue that the seller gets for different realizations of players' valuations are as indicated in the graph below:


Therefore, expected revenue of the seller is :
\begin{eqnarray*}  &amp;&amp;\int_r^\hat{v}\int_r^{v_2} v_1 f(v_1)f(v_2)dv_1dv_2 + \int_r^\hat{v}\int_{v_2}^\hat{v} v_2 f(v_1)f(v_2)dv_1dv_2  \\ &amp;&amp;+\int_0^r\int_r^\hat{v} r f(v_1)f(v_2)dv_1dv_2  + \int_r^\hat{v}\int_0^r r f(v_1)f(v_2)dv_1dv_2 \\ &amp;=&amp;  \int_r^\hat{v}v_1 (1-F(v_1))f(v_1)dv_1 + \int_r^\hat{v} v_2 (1-F(v_2))f(v_2)dv_2 + 2rF(r)(1-F(r)) \\ &amp;=&amp; 2\int_r^\hat{v} v_2 (1-F(v_2))f(v_2)dv_2 + 2rF(r)(1-F(r))\end{eqnarray*}