Total revenue and demand from marginal revenue

Statement

The marginal revenue is $R'(q)=84-4q-q^{2}$ (in €). Find the total revenue function $R(q)$ and the demand function $p=\dfrac{R(q)}{q}$.

Solution

$R(q)=\int (84-4q-q^{2})\,dq=84q-2q^{2}-\frac{1}{3}q^{3}+c.$ Since $R(0)=0$, $c=0$. Demand is the average revenue: $p=\frac{R(q)}{q}=\boxed{84-2q-\frac{1}{3}q^{2}}.$