Producer surplus with quadratic supply

Statement

Supply is $p=(q+2)^{2}$ (in €). At the market price $p_0=64\ \text{€}$ (with $q_0=6$) compute the producer surplus.

Solution

$PS=p_0\,q_0-\displaystyle\int_{0}^{q_0} p(q)\,dq$. $PS=64\cdot 6-\int_{0}^{6}(q+2)^{2}\,dq=384-\left[\frac{(q+2)^{3}}{3}\right]_{0}^{6}.$ $PS=384-\frac{8^{3}-2^{3}}{3}=384-\frac{504}{3}=384-168=\boxed{216\ \text{€}}.$