# Calculus Examples

,

Step 1

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

Step 2

Step 2.1

Multiply by .

Step 2.2

Split the single integral into multiple integrals.

Step 2.3

Apply the constant rule.

Step 2.4

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

Step 2.5

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

Step 2.6

Simplify the answer.

Step 2.6.1

Combine and .

Step 2.6.2

Substitute and simplify.

Step 2.6.2.1

Evaluate at and at .

Step 2.6.2.2

Evaluate at and at .

Step 2.6.2.3

Simplify.

Step 2.6.2.3.1

Multiply by .

Step 2.6.2.3.2

Multiply by .

Step 2.6.2.3.3

Add and .

Step 2.6.2.3.4

Raise to the power of .

Step 2.6.2.3.5

Raising to any positive power yields .

Step 2.6.2.3.6

Cancel the common factor of and .

Step 2.6.2.3.6.1

Factor out of .

Step 2.6.2.3.6.2

Cancel the common factors.

Step 2.6.2.3.6.2.1

Factor out of .

Step 2.6.2.3.6.2.2

Cancel the common factor.

Step 2.6.2.3.6.2.3

Rewrite the expression.

Step 2.6.2.3.6.2.4

Divide by .

Step 2.6.2.3.7

Multiply by .

Step 2.6.2.3.8

Add and .

Step 2.6.2.3.9

Combine and .

Step 2.6.2.3.10

Multiply by .

Step 2.6.2.3.11

Move the negative in front of the fraction.

Step 2.6.2.3.12

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

Step 2.6.2.3.13

Combine and .

Step 2.6.2.3.14

Combine the numerators over the common denominator.

Step 2.6.2.3.15

Simplify the numerator.

Step 2.6.2.3.15.1

Multiply by .

Step 2.6.2.3.15.2

Subtract from .

Step 2.6.2.3.16

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

Step 2.6.2.3.17

Combine and .

Step 2.6.2.3.18

Combine the numerators over the common denominator.

Step 2.6.2.3.19

Simplify the numerator.

Step 2.6.2.3.19.1

Multiply by .

Step 2.6.2.3.19.2

Subtract from .

Step 2.7

Divide by .