# Calculus Examples

,
Step 1
Set up the producer surplus where is the equilibrium quantity and is the equilibrium price.
Step 2
Evaluate the integral and simplify.
Step 2.1
Multiply by .
Step 2.2
Since is constant with respect to , move out of the integral.
Step 2.3
Multiply by .
Step 2.4
By the Power Rule, the integral of with respect to is .
Step 2.5
Step 2.5.1
Combine and .
Step 2.5.2
Substitute and simplify.
Step 2.5.2.1
Evaluate at and at .
Step 2.5.2.2
Simplify.
Step 2.5.2.2.1
Raise to the power of .
Step 2.5.2.2.2
Cancel the common factor of and .
Step 2.5.2.2.2.1
Factor out of .
Step 2.5.2.2.2.2
Cancel the common factors.
Step 2.5.2.2.2.2.1
Factor out of .
Step 2.5.2.2.2.2.2
Cancel the common factor.
Step 2.5.2.2.2.2.3
Rewrite the expression.
Step 2.5.2.2.2.2.4
Divide by .
Step 2.5.2.2.3
Raising to any positive power yields .
Step 2.5.2.2.4
Cancel the common factor of and .
Step 2.5.2.2.4.1
Factor out of .
Step 2.5.2.2.4.2
Cancel the common factors.
Step 2.5.2.2.4.2.1
Factor out of .
Step 2.5.2.2.4.2.2
Cancel the common factor.
Step 2.5.2.2.4.2.3
Rewrite the expression.
Step 2.5.2.2.4.2.4
Divide by .
Step 2.5.2.2.5
Multiply by .
Step 2.5.2.2.6