Brett Boyer: Numerade Educator Profile

Calculus: Early Transcendentals

The supply function $ p_s (x) $ for a commodity gives the relation between the selling price and the number of units that manufacturers will produce at that price. For a higher price, manufacturers will produce more units, so $ p_s $ is an increasing function of $ x $. Let $ X $ be the amount of the commodity currently produced and let $ P = p_s (X) $ be the current price. Some producers would be willing to make and sell the commodity for a lower selling price and are therefore receiving more than their minimal price. The excess is called the producer surplus. An argument similar to that for consumer surplus shows that the surplus is given by the integral
$$ \int_0^X [P - p_s (x)]\ dx $$
Calculate the producer surplus for the supply function $ p_s (x) = 3 + 0.01x^2 $ at the sales level $ X = 10 $. Illustrate by drawing the supply curve and identifying the producer surplus as an area.

Chapter 8: Further Applications of Integration

Section 4: Applications to Economics and Biology