Consumer and producer surplus at equilibrium

Statement

In a market, demand is $p=25-Q$ and supply is $p=2Q+1$ (in €). At equilibrium compute the consumer surplus and the producer surplus.

Solution

Equilibrium: $25-Q=2Q+1\Rightarrow Q^{*}=8,\ P^{*}=17\ \text{€}$. $CS=\int_{0}^{8}(25-Q)\,dQ-17\cdot 8=\big[25Q-\tfrac{Q^{2}}{2}\big]_{0}^{8}-136=168-136=\boxed{32\ \text{€}}.$ $PS=17\cdot 8-\int_{0}^{8}(2Q+1)\,dQ=136-\big[Q^{2}+Q\big]_{0}^{8}=136-72=\boxed{64\ \text{€}}.$