Calculus Examples

$L (x) = x^{4}$

Step 1

Use the Gini Index formula $G = 2 \int_{0}^{1} x - L (x) d x$ .

Step 2

Substitute $x^{4}$ for $L (x)$ .

$G = 2 \int_{0}^{1} x - x^{4} d x$

Step 3

Evaluate $2 \int_{0}^{1} x - x^{4} d x$ .

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Split the single integral into multiple integrals.

$G = 2 (\int_{0}^{1} x d x + \int_{0}^{1} - x^{4} d x)$

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

$G = 2 (\frac{1}{2} x^{2}]_{0}^{1} + \int_{0}^{1} - x^{4} d x)$

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

$G = 2 (\frac{x^{2}}{2}]_{0}^{1} + \int_{0}^{1} - x^{4} d x)$

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$G = 2 (\frac{x^{2}}{2}]_{0}^{1} - \int_{0}^{1} x^{4} d x)$

By the Power Rule, the integral of $x^{4}$ with respect to $x$ is $\frac{1}{5} x^{5}$ .

$G = 2 (\frac{x^{2}}{2}]_{0}^{1} - (\frac{1}{5} x^{5}]_{0}^{1}))$

Simplify the answer.

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Combine $\frac{1}{5}$ and $x^{5}$ .

$G = 2 (\frac{x^{2}}{2}]_{0}^{1} - (\frac{x^{5}}{5}]_{0}^{1}))$

Substitute and simplify.

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Evaluate $\frac{x^{2}}{2}$ at $1$ and at $0$ .

$G = 2 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2} - (\frac{x^{5}}{5}]_{0}^{1}))$

Evaluate $\frac{x^{5}}{5}$ at $1$ and at $0$ .

$G = 2 (\frac{1^{2}}{2} - \frac{0^{2}}{2} - (\frac{1^{5}}{5} - \frac{0^{5}}{5}))$

Simplify.

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One to any power is one.

$G = 2 (\frac{1}{2} - \frac{0^{2}}{2} - (\frac{1^{5}}{5} - \frac{0^{5}}{5}))$

Raising $0$ to any positive power yields $0$ .

$G = 2 (\frac{1}{2} - \frac{0}{2} - (\frac{1^{5}}{5} - \frac{0^{5}}{5}))$

Cancel the common factor of $0$ and $2$ .

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Factor $2$ out of $0$ .

$G = 2 (\frac{1}{2} - \frac{2 (0)}{2} - (\frac{1^{5}}{5} - \frac{0^{5}}{5}))$

Cancel the common factors.

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Factor $2$ out of $2$ .

$G = 2 (\frac{1}{2} - \frac{2 \cdot 0}{2 \cdot 1} - (\frac{1^{5}}{5} - \frac{0^{5}}{5}))$

Cancel the common factor.

$G = 2 (\frac{1}{2} - \frac{2 \cdot 0}{2 \cdot 1} - (\frac{1^{5}}{5} - \frac{0^{5}}{5}))$

Rewrite the expression.

$G = 2 (\frac{1}{2} - \frac{0}{1} - (\frac{1^{5}}{5} - \frac{0^{5}}{5}))$

Divide $0$ by $1$ .

$G = 2 (\frac{1}{2} - 0 - (\frac{1^{5}}{5} - \frac{0^{5}}{5}))$

Multiply $- 1$ by $0$ .

$G = 2 (\frac{1}{2} + 0 - (\frac{1^{5}}{5} - \frac{0^{5}}{5}))$

Add $\frac{1}{2}$ and $0$ .

$G = 2 (\frac{1}{2} - (\frac{1^{5}}{5} - \frac{0^{5}}{5}))$

One to any power is one.

$G = 2 (\frac{1}{2} - (\frac{1}{5} - \frac{0^{5}}{5}))$

Raising $0$ to any positive power yields $0$ .

$G = 2 (\frac{1}{2} - (\frac{1}{5} - \frac{0}{5}))$

Cancel the common factor of $0$ and $5$ .

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Factor $5$ out of $0$ .

$G = 2 (\frac{1}{2} - (\frac{1}{5} - \frac{5 (0)}{5}))$

Cancel the common factors.

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Factor $5$ out of $5$ .

$G = 2 (\frac{1}{2} - (\frac{1}{5} - \frac{5 \cdot 0}{5 \cdot 1}))$

Cancel the common factor.

$G = 2 (\frac{1}{2} - (\frac{1}{5} - \frac{5 \cdot 0}{5 \cdot 1}))$

Rewrite the expression.

$G = 2 (\frac{1}{2} - (\frac{1}{5} - \frac{0}{1}))$

Divide $0$ by $1$ .

$G = 2 (\frac{1}{2} - (\frac{1}{5} - 0))$

Multiply $- 1$ by $0$ .

$G = 2 (\frac{1}{2} - (\frac{1}{5} + 0))$

Add $\frac{1}{5}$ and $0$ .

$G = 2 (\frac{1}{2} - \frac{1}{5})$

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$G = 2 (\frac{1}{2} \cdot \frac{5}{5} - \frac{1}{5})$

To write $- \frac{1}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$G = 2 (\frac{1}{2} \cdot \frac{5}{5} - \frac{1}{5} \cdot \frac{2}{2})$

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

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Multiply $\frac{1}{2}$ by $\frac{5}{5}$ .

$G = 2 (\frac{5}{2 \cdot 5} - \frac{1}{5} \cdot \frac{2}{2})$

Multiply $2$ by $5$ .

$G = 2 (\frac{5}{10} - \frac{1}{5} \cdot \frac{2}{2})$

Multiply $\frac{1}{5}$ by $\frac{2}{2}$ .

$G = 2 (\frac{5}{10} - \frac{2}{5 \cdot 2})$

Multiply $5$ by $2$ .

$G = 2 (\frac{5}{10} - \frac{2}{10})$

Combine the numerators over the common denominator.

$G = 2 \frac{5 - 2}{10}$

Subtract $2$ from $5$ .

$G = 2 (\frac{3}{10})$

Combine $2$ and $\frac{3}{10}$ .

$G = \frac{2 \cdot 3}{10}$

Multiply $2$ by $3$ .

$G = \frac{6}{10}$

Cancel the common factor of $6$ and $10$ .

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Factor $2$ out of $6$ .

$G = \frac{2 (3)}{10}$

Cancel the common factors.

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Factor $2$ out of $10$ .

$G = \frac{2 \cdot 3}{2 \cdot 5}$

Cancel the common factor.

$G = \frac{2 \cdot 3}{2 \cdot 5}$

Rewrite the expression.

$G = \frac{3}{5}$

Step 4

Convert to decimal.

$G = 0.6$

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