Q:  A monopolist with TC=3q²+q+12 faces a demand curve of P=81-2q. 
        Find CS and DWL.

A: First we need to determine Pm and Qm.  From the information above we know:
    MC=6q+1 and MR=81-4q.  To find qm set MC=MR and you get 10q=80 or qm=8. 
    Pm is found by plugging qm into the demand curve:  pm=81-2(8)=65.
    To find CS we still need to know the P-intercept of demand which is P=81-2(0)=81.
    Hence CS= (1/2) (81-65)(8)=64

    To find DWL we need to know qe and the height of MC at qm. 
    To find qe set P=MC or 81-2q=6q+1 or qe=10.
    At qm the height of MC is 6(8)+1=49.
    Hence DWL= (1/2)(65-49)*(10-8)=16.