# Calculus Examples

,

Step 1

Set up the producer surplus where is the equilibrium quantity and is the equilibrium price.

Step 2

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

Simplify the answer.

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

Add and .

Step 2.5.2.2.7

Multiply by .

Step 2.5.2.2.8

Subtract from .