# Calculus Examples

Step 1
Use the Gini Index formula .
Step 2
Substitute for .
Step 3
Evaluate .
Step 3.1
Split the single integral into multiple integrals.
Step 3.2
By the Power Rule, the integral of with respect to is .
Step 3.3
Combine and .
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
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
One to any power is one.
Step 3.6.2.3.2
Raising to any positive power yields .
Step 3.6.2.3.3
Cancel the common factor of and .
Step 3.6.2.3.3.1
Factor out of .
Step 3.6.2.3.3.2
Cancel the common factors.
Step 3.6.2.3.3.2.1
Factor out of .
Step 3.6.2.3.3.2.2
Cancel the common factor.
Step 3.6.2.3.3.2.3
Rewrite the expression.
Step 3.6.2.3.3.2.4
Divide by .
Step 3.6.2.3.4
Multiply by .
Step 3.6.2.3.5
Step 3.6.2.3.6
One to any power is one.
Step 3.6.2.3.7
Raising to any positive power yields .
Step 3.6.2.3.8
Cancel the common factor of and .
Step 3.6.2.3.8.1
Factor out of .
Step 3.6.2.3.8.2
Cancel the common factors.
Step 3.6.2.3.8.2.1
Factor out of .
Step 3.6.2.3.8.2.2
Cancel the common factor.
Step 3.6.2.3.8.2.3
Rewrite the expression.
Step 3.6.2.3.8.2.4
Divide by .
Step 3.6.2.3.9
Multiply by .
Step 3.6.2.3.10
Step 3.6.2.3.11
To write as a fraction with a common denominator, multiply by .
Step 3.6.2.3.12
To write as a fraction with a common denominator, multiply by .
Step 3.6.2.3.13
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Step 3.6.2.3.13.1
Multiply by .
Step 3.6.2.3.13.2
Multiply by .
Step 3.6.2.3.13.3
Multiply by .
Step 3.6.2.3.13.4
Multiply by .
Step 3.6.2.3.14
Combine the numerators over the common denominator.
Step 3.6.2.3.15
Subtract from .
Step 3.6.2.3.16
Combine and .
Step 3.6.2.3.17
Cancel the common factor of and .
Step 3.6.2.3.17.1
Factor out of .
Step 3.6.2.3.17.2
Cancel the common factors.
Step 3.6.2.3.17.2.1
Factor out of .
Step 3.6.2.3.17.2.2
Cancel the common factor.
Step 3.6.2.3.17.2.3
Rewrite the expression.
Step 4
Convert to decimal.