# Calculus Examples

,

Step 1

Step 1.1

Find the equilibrium quantity.

Step 1.1.1

Find the equilibrium point by setting the supply function equal to the demand function.

Step 1.1.2

Solve .

Step 1.1.2.1

Subtract from both sides of the equation.

Step 1.1.2.2

Factor the left side of the equation.

Step 1.1.2.2.1

Factor out of .

Step 1.1.2.2.1.1

Reorder the expression.

Step 1.1.2.2.1.1.1

Move .

Step 1.1.2.2.1.1.2

Reorder and .

Step 1.1.2.2.1.2

Factor out of .

Step 1.1.2.2.1.3

Factor out of .

Step 1.1.2.2.1.4

Rewrite as .

Step 1.1.2.2.1.5

Factor out of .

Step 1.1.2.2.1.6

Factor out of .

Step 1.1.2.2.2

Factor.

Step 1.1.2.2.2.1

Factor using the AC method.

Step 1.1.2.2.2.1.1

Consider the form . Find a pair of integers whose product is and whose sum is . In this case, whose product is and whose sum is .

Step 1.1.2.2.2.1.2

Write the factored form using these integers.

Step 1.1.2.2.2.2

Remove unnecessary parentheses.

Step 1.1.2.3

If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .

Step 1.1.2.4

Set equal to and solve for .

Step 1.1.2.4.1

Set equal to .

Step 1.1.2.4.2

Add to both sides of the equation.

Step 1.1.2.5

Set equal to and solve for .

Step 1.1.2.5.1

Set equal to .

Step 1.1.2.5.2

Subtract from both sides of the equation.

Step 1.1.2.6

The final solution is all the values that make true.

Step 1.1.3

Ignore the negative solution.

Step 1.2

Find the equilibrium price.

Step 1.2.1

Find the equilibrium price by substituting the equilibrium quantity for in .

Step 1.2.2

Simplify .

Step 1.2.2.1

Multiply by .

Step 1.2.2.2

Subtract from .

Step 1.3

Write the equilibrium point.

Step 2

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

Step 3

Step 3.1

Multiply by .

Step 3.2

Split the single integral into multiple integrals.

Step 3.3

Apply the constant rule.

Step 3.4

Since is constant with respect to , move out of the integral.

Step 3.5

By the Power Rule, the integral of with respect to is .

Step 3.6

Simplify the answer.

Step 3.6.1

Combine and .

Step 3.6.2

Substitute and simplify.

Step 3.6.2.1

Evaluate at and at .

Step 3.6.2.2

Evaluate at and at .

Step 3.6.2.3

Simplify.

Step 3.6.2.3.1

Multiply by .

Step 3.6.2.3.2

Multiply by .

Step 3.6.2.3.3

Add and .

Step 3.6.2.3.4

Raise to the power of .

Step 3.6.2.3.5

Raising to any positive power yields .

Step 3.6.2.3.6

Cancel the common factor of and .

Step 3.6.2.3.6.1

Factor out of .

Step 3.6.2.3.6.2

Cancel the common factors.

Step 3.6.2.3.6.2.1

Factor out of .

Step 3.6.2.3.6.2.2

Cancel the common factor.

Step 3.6.2.3.6.2.3

Rewrite the expression.

Step 3.6.2.3.6.2.4

Divide by .

Step 3.6.2.3.7

Multiply by .

Step 3.6.2.3.8

Add and .

Step 3.6.2.3.9

Combine and .

Step 3.6.2.3.10

Multiply by .

Step 3.6.2.3.11

Move the negative in front of the fraction.

Step 3.6.2.3.12

To write as a fraction with a common denominator, multiply by .

Step 3.6.2.3.13

Combine and .

Step 3.6.2.3.14

Combine the numerators over the common denominator.

Step 3.6.2.3.15

Simplify the numerator.

Step 3.6.2.3.15.1

Multiply by .

Step 3.6.2.3.15.2

Subtract from .

Step 3.6.2.3.16

To write as a fraction with a common denominator, multiply by .

Step 3.6.2.3.17

Combine and .

Step 3.6.2.3.18

Combine the numerators over the common denominator.

Step 3.6.2.3.19

Simplify the numerator.

Step 3.6.2.3.19.1

Multiply by .

Step 3.6.2.3.19.2

Subtract from .

Step 3.7

Divide by .