Calculus Examples

L (x) = x^{2}

$L (x) = x^{2}$

Step 1

Use the Gini Index formula

G = 2 \int_{0}^{1} x - L (x) d x

$G = 2 \int_{0}^{1} x - L (x) d x$ .

Step 2

Substitute

x^{2}

$x^{2}$ for

L (x)

$L (x)$ .

G = 2 \int_{0}^{1} x - x^{2} d x

$G = 2 \int_{0}^{1} x - x^{2} d x$

Step 3

Evaluate

2 \int_{0}^{1} x - x^{2} d x

$2 \int_{0}^{1} x - x^{2} d x$ .

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Step 3.1

Split the single integral into multiple integrals.

G = 2 (\int_{0}^{1} x d x + \int_{0}^{1} - x^{2} d x)

$G = 2 (\int_{0}^{1} x d x + \int_{0}^{1} - x^{2} d x)$

Step 3.2

By the Power Rule, the integral of

x

$x$ with respect to

x

$x$ is

\frac{1}{2} x^{2}

$\frac{1}{2} x^{2}$ .

G = 2 (\frac{1}{2} x^{2}]_{0}^{1} + \int_{0}^{1} - x^{2} d x)

$G = 2 (\frac{1}{2} x^{2}]_{0}^{1} + \int_{0}^{1} - x^{2} d x)$

Step 3.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

x^{2}

$x^{2}$ .

G = 2 (\frac{x^{2}}{2}]_{0}^{1} + \int_{0}^{1} - x^{2} d x)

$G = 2 (\frac{x^{2}}{2}]_{0}^{1} + \int_{0}^{1} - x^{2} d x)$

Step 3.4

Since

- 1

$- 1$ is constant with respect to

x

$x$ , move

- 1

$- 1$ out of the integral.

G = 2 (\frac{x^{2}}{2}]_{0}^{1} - \int_{0}^{1} x^{2} d x)

$G = 2 (\frac{x^{2}}{2}]_{0}^{1} - \int_{0}^{1} x^{2} d x)$

Step 3.5

By the Power Rule, the integral of

x^{2}

$x^{2}$ with respect to

x

$x$ is

\frac{1}{3} x^{3}

$\frac{1}{3} x^{3}$ .

G = 2 (\frac{x^{2}}{2}]_{0}^{1} - (\frac{1}{3} x^{3}]_{0}^{1}))

$G = 2 (\frac{x^{2}}{2}]_{0}^{1} - (\frac{1}{3} x^{3}]_{0}^{1}))$

Step 3.6

Simplify the answer.

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Step 3.6.1

Combine

\frac{1}{3}

$\frac{1}{3}$ and

x^{3}

$x^{3}$ .

G = 2 (\frac{x^{2}}{2}]_{0}^{1} - (\frac{x^{3}}{3}]_{0}^{1}))

$G = 2 (\frac{x^{2}}{2}]_{0}^{1} - (\frac{x^{3}}{3}]_{0}^{1}))$

Step 3.6.2

Substitute and simplify.

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Step 3.6.2.1

Evaluate

\frac{x^{2}}{2}

$\frac{x^{2}}{2}$ at

1

$1$ and at

0

$0$ .

G = 2 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2} - (\frac{x^{3}}{3}]_{0}^{1}))

$G = 2 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2} - (\frac{x^{3}}{3}]_{0}^{1}))$

Step 3.6.2.2

Evaluate

\frac{x^{3}}{3}

$\frac{x^{3}}{3}$ at

1

$1$ and at

0

$0$ .

G = 2 (\frac{1^{2}}{2} - \frac{0^{2}}{2} - (\frac{1^{3}}{3} - \frac{0^{3}}{3}))

$G = 2 (\frac{1^{2}}{2} - \frac{0^{2}}{2} - (\frac{1^{3}}{3} - \frac{0^{3}}{3}))$

Step 3.6.2.3

Simplify.

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Step 3.6.2.3.1

One to any power is one.

G = 2 (\frac{1}{2} - \frac{0^{2}}{2} - (\frac{1^{3}}{3} - \frac{0^{3}}{3}))

$G = 2 (\frac{1}{2} - \frac{0^{2}}{2} - (\frac{1^{3}}{3} - \frac{0^{3}}{3}))$

Step 3.6.2.3.2

Raising

0

$0$ to any positive power yields

0

$0$ .

G = 2 (\frac{1}{2} - \frac{0}{2} - (\frac{1^{3}}{3} - \frac{0^{3}}{3}))

$G = 2 (\frac{1}{2} - \frac{0}{2} - (\frac{1^{3}}{3} - \frac{0^{3}}{3}))$

Step 3.6.2.3.3

Cancel the common factor of

0

$0$ and

2

$2$ .

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Step 3.6.2.3.3.1

Factor

2

$2$ out of

0

$0$ .

G = 2 (\frac{1}{2} - \frac{2 (0)}{2} - (\frac{1^{3}}{3} - \frac{0^{3}}{3}))

$G = 2 (\frac{1}{2} - \frac{2 (0)}{2} - (\frac{1^{3}}{3} - \frac{0^{3}}{3}))$

Step 3.6.2.3.3.2

Cancel the common factors.

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Step 3.6.2.3.3.2.1

Factor

2

$2$ out of

2

$2$ .

G = 2 (\frac{1}{2} - \frac{2 \cdot 0}{2 \cdot 1} - (\frac{1^{3}}{3} - \frac{0^{3}}{3}))

$G = 2 (\frac{1}{2} - \frac{2 \cdot 0}{2 \cdot 1} - (\frac{1^{3}}{3} - \frac{0^{3}}{3}))$

Step 3.6.2.3.3.2.2

Cancel the common factor.

$G = 2 (\frac{1}{2} - \frac{2 \cdot 0}{2 \cdot 1} - (\frac{1^{3}}{3} - \frac{0^{3}}{3}))$

Step 3.6.2.3.3.2.3

Rewrite the expression.

$G = 2 (\frac{1}{2} - \frac{0}{1} - (\frac{1^{3}}{3} - \frac{0^{3}}{3}))$

Step 3.6.2.3.3.2.4

Divide

$0$ by

$1$ .

$G = 2 (\frac{1}{2} - 0 - (\frac{1^{3}}{3} - \frac{0^{3}}{3}))$

Step 3.6.2.3.4

Multiply

$- 1$ by

$0$ .

$G = 2 (\frac{1}{2} + 0 - (\frac{1^{3}}{3} - \frac{0^{3}}{3}))$

Step 3.6.2.3.5

Add

$\frac{1}{2}$ and

$0$ .

$G = 2 (\frac{1}{2} - (\frac{1^{3}}{3} - \frac{0^{3}}{3}))$

Step 3.6.2.3.6

One to any power is one.

$G = 2 (\frac{1}{2} - (\frac{1}{3} - \frac{0^{3}}{3}))$

Step 3.6.2.3.7

Raising

$0$ to any positive power yields

$0$ .

$G = 2 (\frac{1}{2} - (\frac{1}{3} - \frac{0}{3}))$

Step 3.6.2.3.8

Cancel the common factor of

$0$ and

$3$ .

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Step 3.6.2.3.8.1

Factor

$3$ out of

$0$ .

$G = 2 (\frac{1}{2} - (\frac{1}{3} - \frac{3 (0)}{3}))$

Step 3.6.2.3.8.2

Cancel the common factors.

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Step 3.6.2.3.8.2.1

Factor

$3$ out of

$3$ .

$G = 2 (\frac{1}{2} - (\frac{1}{3} - \frac{3 \cdot 0}{3 \cdot 1}))$

Step 3.6.2.3.8.2.2

Cancel the common factor.

$G = 2 (\frac{1}{2} - (\frac{1}{3} - \frac{3 \cdot 0}{3 \cdot 1}))$

Step 3.6.2.3.8.2.3

Rewrite the expression.

$G = 2 (\frac{1}{2} - (\frac{1}{3} - \frac{0}{1}))$

Step 3.6.2.3.8.2.4

Divide

$0$ by

$1$ .

$G = 2 (\frac{1}{2} - (\frac{1}{3} - 0))$

Step 3.6.2.3.9

Multiply

$- 1$ by

$0$ .

$G = 2 (\frac{1}{2} - (\frac{1}{3} + 0))$

Step 3.6.2.3.10

Add

$\frac{1}{3}$ and

$0$ .

$G = 2 (\frac{1}{2} - \frac{1}{3})$

Step 3.6.2.3.11

To write

$\frac{1}{2}$ as a fraction with a common denominator, multiply by

$\frac{3}{3}$ .

$G = 2 (\frac{1}{2} \cdot \frac{3}{3} - \frac{1}{3})$

Step 3.6.2.3.12

To write

$- \frac{1}{3}$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

$G = 2 (\frac{1}{2} \cdot \frac{3}{3} - \frac{1}{3} \cdot \frac{2}{2})$

Step 3.6.2.3.13

Write each expression with a common denominator of

$6$ , by multiplying each by an appropriate factor of

$1$ .

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Step 3.6.2.3.13.1

Multiply

$\frac{1}{2}$ by

$\frac{3}{3}$ .

$G = 2 (\frac{3}{2 \cdot 3} - \frac{1}{3} \cdot \frac{2}{2})$

Step 3.6.2.3.13.2

Multiply

$2$ by

$3$ .

$G = 2 (\frac{3}{6} - \frac{1}{3} \cdot \frac{2}{2})$

Step 3.6.2.3.13.3

Multiply

$\frac{1}{3}$ by

$\frac{2}{2}$ .

$G = 2 (\frac{3}{6} - \frac{2}{3 \cdot 2})$

Step 3.6.2.3.13.4

Multiply

$3$ by

$2$ .

$G = 2 (\frac{3}{6} - \frac{2}{6})$

Step 3.6.2.3.14

Combine the numerators over the common denominator.

$G = 2 \frac{3 - 2}{6}$

Step 3.6.2.3.15

Subtract

$2$ from

$3$ .

$G = 2 (\frac{1}{6})$

Step 3.6.2.3.16

Combine

$2$ and

$\frac{1}{6}$ .

$G = \frac{2}{6}$

Step 3.6.2.3.17

Cancel the common factor of

$2$ and

$6$ .

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Step 3.6.2.3.17.1

Factor

$2$ out of

$2$ .

$G = \frac{2 (1)}{6}$

Step 3.6.2.3.17.2

Cancel the common factors.

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Step 3.6.2.3.17.2.1

Factor

$2$ out of

$6$ .

$G = \frac{2 \cdot 1}{2 \cdot 3}$

Step 3.6.2.3.17.2.2

Cancel the common factor.

$G = \frac{2 \cdot 1}{2 \cdot 3}$

Step 3.6.2.3.17.2.3

Rewrite the expression.

$G = \frac{1}{3}$

Step 4

Convert to decimal.

$G = 0.\overline{3}$

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