# Calculus Examples

,
Step 1
Set up the consumer 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
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
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
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
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 .