# Calculus Examples

,
Step 1
Find the equilibrium point.
Find the equilibrium quantity.
Find the equilibrium point by setting the supply function equal to the demand function.
Solve .
Move all terms containing to the left side of the equation.
Subtract from both sides of the equation.
Subtract from .
Subtract from both sides of the equation.
Divide each term in by and simplify.
Divide each term in by .
Simplify the left side.
Cancel the common factor of .
Cancel the common factor.
Divide by .
Simplify the right side.
Divide by .
Find the equilibrium price.
Find the equilibrium price by substituting the equilibrium quantity for in .
Simplify .
Multiply by .
Write the equilibrium point.
Step 2
Set up the producer surplus where is the equilibrium quantity and is the equilibrium price.
Step 3
Evaluate the integral and simplify.
Multiply by .
Since is constant with respect to , move out of the integral.
Multiply by .
By the Power Rule, the integral of with respect to is .
Combine and .
Substitute and simplify.
Evaluate at and at .
Simplify.
Raise to the power of .
Raising to any positive power yields .
Cancel the common factor of and .
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by .
Multiply by .
Combine and .
Multiply by .
Cancel the common factor of and .
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by .
Subtract from .