6. Integration
6.7: Additional Applications of the Integral
Consumers’ Surplus and Producers’ Surplus
Let p = D(x) be the demand equation, where p is the unit price of a commodity and x is the quantity
demanded by the consumers at that price. Let p = S(x) be the supply equation, where p is the unit
price of a commodity and x is the quantity made available by the producers at that price.
Consumers’ Surplus: The total savings to consumers who are willing to pay more than market price
for a product, but are still able to buy the product for market price.
Z
x0
[D(x) − p0 ] dx.
0
Example 1. Let p = D(x) = 200 − 0.04x, and p0 = 190. Find the consumers’ surplus.
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Example 2. Let p = D(x) = 25 − 2x1/3 , and x0 = 125. Find the consumers’ surplus.
Producers’ Surplus: The total gain to producers who are willing to supply units at a lower price
than market price, but are still able to supply units at market price.
Z
x0
[p0 − S(x)] dx.
0
Example
3. Find the producers’ surplus at a price level of $231 if the supply equation is given by
√
3
p = 3 x + 150.
2
Example 4. If the demand and supply are given by D(x) = 30 − x2 and S(x) = 3x + 15, respectively,
find
a) Find the equilibrium point.
b) The consumers’ surplus at equilibrium price.
c) The producers’ surplus at equilibrium price.
3