Consumer surplus with linear demand

Statement

The demand function is $p=45-0{.}5q$ (in €). At the market price $p_0=30\ \text{€}$ (with $q_0=30$) compute the consumer surplus.

Solution

$CS=\displaystyle\int_{0}^{q_0} p(q)\,dq-p_0\,q_0$. $CS=\int_{0}^{30}(45-0{.}5q)\,dq-30\cdot 30=\big[45q-0{.}25q^{2}\big]_{0}^{30}-900.$ $CS=(1350-225)-900=\boxed{225\ \text{€}}.$