Calculus Examples

$y = x^{3}$ , $y = 4 x$

Step 1

Solve by substitution to find the intersection between the curves.

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Step 1.1

Eliminate the equal sides of each equation and combine.

$x^{3} = 4 x$

Step 1.2

Solve $x^{3} = 4 x$ for $x$ .

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Step 1.2.1

Subtract $4 x$ from both sides of the equation.

$x^{3} - 4 x = 0$

Step 1.2.2

Factor the left side of the equation.

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Step 1.2.2.1

Factor $x$ out of $x^{3} - 4 x$ .

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Step 1.2.2.1.1

Factor $x$ out of $x^{3}$ .

$x \cdot x^{2} - 4 x = 0$

Step 1.2.2.1.2

Factor $x$ out of $- 4 x$ .

$x \cdot x^{2} + x \cdot - 4 = 0$

Step 1.2.2.1.3

Factor $x$ out of $x \cdot x^{2} + x \cdot - 4$ .

$x (x^{2} - 4) = 0$

Step 1.2.2.2

Rewrite $4$ as $2^{2}$ .

$x (x^{2} - 2^{2}) = 0$

Step 1.2.2.3

Factor.

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Step 1.2.2.3.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

$x ((x + 2) (x - 2)) = 0$

Step 1.2.2.3.2

Remove unnecessary parentheses.

$x (x + 2) (x - 2) = 0$

Step 1.2.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x + 2 = 0$

$x - 2 = 0 + y = 4 x$

Step 1.2.4

Set $x$ equal to $0$ .

$x = 0$

Step 1.2.5

Set $x + 2$ equal to $0$ and solve for $x$ .

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Step 1.2.5.1

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 1.2.5.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 1.2.6

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 1.2.6.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 1.2.6.2

Add $2$ to both sides of the equation.

$x = 2$

Step 1.2.7

The final solution is all the values that make $x (x + 2) (x - 2) = 0$ true.

$x = 0, - 2, 2$

Step 1.3

Evaluate $y$ when $x = 0$ .

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Step 1.3.1

Substitute $0$ for $x$ .

$y = 4 (0)$

Step 1.3.2

Multiply $4$ by $0$ .

$y = 0$

Step 1.4

Evaluate $y$ when $x = - 2$ .

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Step 1.4.1

Substitute $- 2$ for $x$ .

$y = 4 (- 2)$

Step 1.4.2

Multiply $4$ by $- 2$ .

$y = - 8$

Step 1.5

Evaluate $y$ when $x = 2$ .

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Step 1.5.1

Substitute $2$ for $x$ .

$y = 4 (2)$

Step 1.5.2

Multiply $4$ by $2$ .

$y = 8$

Step 1.6

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(0, 0)$

$(- 2, - 8)$

$(2, 8)$

$(0, 0)$

$(- 2, - 8)$

$(2, 8)$

Step 2

The area of the region between the curves is defined as the integral of the upper curve minus the integral of the lower curve over each region. The regions are determined by the intersection points of the curves. This can be done algebraically or graphically.

$A r e a = \int_{- 2}^{0} x^{3} d x - \int_{- 2}^{0} 4 x d x$

Step 3

Integrate to find the area between $- 2$ and $0$ .

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Step 3.1

Combine the integrals into a single integral.

$\int_{- 2}^{0} x^{3} - (4 x) d x$

Step 3.2

Multiply $4$ by $- 1$ .

$\int_{- 2}^{0} x^{3} - 4 x d x$

Step 3.3

Split the single integral into multiple integrals.

$\int_{- 2}^{0} x^{3} d x + \int_{- 2}^{0} - 4 x d x$

Step 3.4

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

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

Step 3.5

Since $- 4$ is constant with respect to $x$ , move $- 4$ out of the integral.

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

Step 3.6

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

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

Step 3.7

Simplify the answer.

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Step 3.7.1

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

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

Step 3.7.2

Substitute and simplify.

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Step 3.7.2.1

Evaluate $\frac{1}{4} x^{4}$ at $0$ and at $- 2$ .

$(\frac{1}{4} \cdot 0^{4}) - \frac{1}{4} {(- 2)}^{4} - 4 (\frac{x^{2}}{2}]_{- 2}^{0})$

Step 3.7.2.2

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

$\frac{1}{4} \cdot 0^{4} - \frac{1}{4} {(- 2)}^{4} - 4 (\frac{0^{2}}{2} - \frac{{(- 2)}^{2}}{2})$

Step 3.7.2.3

Simplify.

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Step 3.7.2.3.1

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

$\frac{1}{4} \cdot 0 - \frac{1}{4} {(- 2)}^{4} - 4 (\frac{0^{2}}{2} - \frac{{(- 2)}^{2}}{2})$

Step 3.7.2.3.2

Multiply $\frac{1}{4}$ by $0$ .

$0 - \frac{1}{4} {(- 2)}^{4} - 4 (\frac{0^{2}}{2} - \frac{{(- 2)}^{2}}{2})$

Step 3.7.2.3.3

Raise $- 2$ to the power of $4$ .

$0 - \frac{1}{4} \cdot 16 - 4 (\frac{0^{2}}{2} - \frac{{(- 2)}^{2}}{2})$

Step 3.7.2.3.4

Multiply $16$ by $- 1$ .

$0 - 16 (\frac{1}{4}) - 4 (\frac{0^{2}}{2} - \frac{{(- 2)}^{2}}{2})$

Step 3.7.2.3.5

Combine $- 16$ and $\frac{1}{4}$ .

$0 + \frac{- 16}{4} - 4 (\frac{0^{2}}{2} - \frac{{(- 2)}^{2}}{2})$

Step 3.7.2.3.6

Cancel the common factor of $- 16$ and $4$ .

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Step 3.7.2.3.6.1

Factor $4$ out of $- 16$ .

$0 + \frac{4 \cdot - 4}{4} - 4 (\frac{0^{2}}{2} - \frac{{(- 2)}^{2}}{2})$

Step 3.7.2.3.6.2

Cancel the common factors.

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Step 3.7.2.3.6.2.1

Factor $4$ out of $4$ .

$0 + \frac{4 \cdot - 4}{4 (1)} - 4 (\frac{0^{2}}{2} - \frac{{(- 2)}^{2}}{2})$

Step 3.7.2.3.6.2.2

Cancel the common factor.

$0 + \frac{4 \cdot - 4}{4 \cdot 1} - 4 (\frac{0^{2}}{2} - \frac{{(- 2)}^{2}}{2})$

Step 3.7.2.3.6.2.3

Rewrite the expression.

$0 + \frac{- 4}{1} - 4 (\frac{0^{2}}{2} - \frac{{(- 2)}^{2}}{2})$

Step 3.7.2.3.6.2.4

Divide $- 4$ by $1$ .

$0 - 4 - 4 (\frac{0^{2}}{2} - \frac{{(- 2)}^{2}}{2})$

Step 3.7.2.3.7

Subtract $4$ from $0$ .

$- 4 - 4 (\frac{0^{2}}{2} - \frac{{(- 2)}^{2}}{2})$

Step 3.7.2.3.8

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

$- 4 - 4 (\frac{0}{2} - \frac{{(- 2)}^{2}}{2})$

Step 3.7.2.3.9

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

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Step 3.7.2.3.9.1

Factor $2$ out of $0$ .

$- 4 - 4 (\frac{2 (0)}{2} - \frac{{(- 2)}^{2}}{2})$

Step 3.7.2.3.9.2

Cancel the common factors.

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Step 3.7.2.3.9.2.1

Factor $2$ out of $2$ .

$- 4 - 4 (\frac{2 \cdot 0}{2 \cdot 1} - \frac{{(- 2)}^{2}}{2})$

Step 3.7.2.3.9.2.2

Cancel the common factor.

$- 4 - 4 (\frac{2 \cdot 0}{2 \cdot 1} - \frac{{(- 2)}^{2}}{2})$

Step 3.7.2.3.9.2.3

Rewrite the expression.

$- 4 - 4 (\frac{0}{1} - \frac{{(- 2)}^{2}}{2})$

Step 3.7.2.3.9.2.4

Divide $0$ by $1$ .

$- 4 - 4 (0 - \frac{{(- 2)}^{2}}{2})$

Step 3.7.2.3.10

Raise $- 2$ to the power of $2$ .

$- 4 - 4 (0 - \frac{4}{2})$

Step 3.7.2.3.11

Cancel the common factor of $4$ and $2$ .

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Step 3.7.2.3.11.1

Factor $2$ out of $4$ .

$- 4 - 4 (0 - \frac{2 \cdot 2}{2})$

Step 3.7.2.3.11.2

Cancel the common factors.

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Step 3.7.2.3.11.2.1

Factor $2$ out of $2$ .

$- 4 - 4 (0 - \frac{2 \cdot 2}{2 (1)})$

Step 3.7.2.3.11.2.2

Cancel the common factor.

$- 4 - 4 (0 - \frac{2 \cdot 2}{2 \cdot 1})$

Step 3.7.2.3.11.2.3

Rewrite the expression.

$- 4 - 4 (0 - \frac{2}{1})$

Step 3.7.2.3.11.2.4

Divide $2$ by $1$ .

$- 4 - 4 (0 - 1 \cdot 2)$

Step 3.7.2.3.12

Multiply $- 1$ by $2$ .

$- 4 - 4 (0 - 2)$

Step 3.7.2.3.13

Subtract $2$ from $0$ .

$- 4 - 4 \cdot - 2$

Step 3.7.2.3.14

Multiply $- 4$ by $- 2$ .

$- 4 + 8$

Step 3.7.2.3.15

Add $- 4$ and $8$ .

$4$

Step 4

$A r e a = \int_{0}^{2} 4 x d x - \int_{0}^{2} x^{3} d x$

Step 5

Integrate to find the area between $0$ and $2$ .

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Step 5.1

Combine the integrals into a single integral.

$\int_{0}^{2} 4 x - (x^{3}) d x$

Step 5.2

Multiply $- 1$ by $x^{3}$ .

$\int_{0}^{2} 4 x - x^{3} d x$

Step 5.3

Split the single integral into multiple integrals.

$\int_{0}^{2} 4 x d x + \int_{0}^{2} - x^{3} d x$

Step 5.4

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$4 \int_{0}^{2} x d x + \int_{0}^{2} - x^{3} d x$

Step 5.5

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

$4 (\frac{1}{2} x^{2}]_{0}^{2}) + \int_{0}^{2} - x^{3} d x$

Step 5.6

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

$4 (\frac{x^{2}}{2}]_{0}^{2}) + \int_{0}^{2} - x^{3} d x$

Step 5.7

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

$4 (\frac{x^{2}}{2}]_{0}^{2}) - \int_{0}^{2} x^{3} d x$

Step 5.8

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

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

Step 5.9

Simplify the answer.

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Step 5.9.1

Combine $\frac{1}{4}$ and $x^{4}$ .

$4 (\frac{x^{2}}{2}]_{0}^{2}) - (\frac{x^{4}}{4}]_{0}^{2})$

Step 5.9.2

Substitute and simplify.

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Step 5.9.2.1

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

$4 ((\frac{2^{2}}{2}) - \frac{0^{2}}{2}) - (\frac{x^{4}}{4}]_{0}^{2})$

Step 5.9.2.2

Evaluate $\frac{x^{4}}{4}$ at $2$ and at $0$ .

$4 (\frac{2^{2}}{2} - \frac{0^{2}}{2}) - (\frac{2^{4}}{4} - \frac{0^{4}}{4})$

Step 5.9.2.3

Simplify.

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Step 5.9.2.3.1

Raise $2$ to the power of $2$ .

$4 (\frac{4}{2} - \frac{0^{2}}{2}) - (\frac{2^{4}}{4} - \frac{0^{4}}{4})$

Step 5.9.2.3.2

Cancel the common factor of $4$ and $2$ .

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Step 5.9.2.3.2.1

Factor $2$ out of $4$ .

$4 (\frac{2 \cdot 2}{2} - \frac{0^{2}}{2}) - (\frac{2^{4}}{4} - \frac{0^{4}}{4})$

Step 5.9.2.3.2.2

Cancel the common factors.

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Step 5.9.2.3.2.2.1

Factor $2$ out of $2$ .

$4 (\frac{2 \cdot 2}{2 (1)} - \frac{0^{2}}{2}) - (\frac{2^{4}}{4} - \frac{0^{4}}{4})$

Step 5.9.2.3.2.2.2

Cancel the common factor.

$4 (\frac{2 \cdot 2}{2 \cdot 1} - \frac{0^{2}}{2}) - (\frac{2^{4}}{4} - \frac{0^{4}}{4})$

Step 5.9.2.3.2.2.3

Rewrite the expression.

$4 (\frac{2}{1} - \frac{0^{2}}{2}) - (\frac{2^{4}}{4} - \frac{0^{4}}{4})$

Step 5.9.2.3.2.2.4

Divide $2$ by $1$ .

$4 (2 - \frac{0^{2}}{2}) - (\frac{2^{4}}{4} - \frac{0^{4}}{4})$

Step 5.9.2.3.3

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

$4 (2 - \frac{0}{2}) - (\frac{2^{4}}{4} - \frac{0^{4}}{4})$

Step 5.9.2.3.4

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

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Step 5.9.2.3.4.1

Factor $2$ out of $0$ .

$4 (2 - \frac{2 (0)}{2}) - (\frac{2^{4}}{4} - \frac{0^{4}}{4})$

Step 5.9.2.3.4.2

Cancel the common factors.

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Step 5.9.2.3.4.2.1

Factor $2$ out of $2$ .

$4 (2 - \frac{2 \cdot 0}{2 \cdot 1}) - (\frac{2^{4}}{4} - \frac{0^{4}}{4})$

Step 5.9.2.3.4.2.2

Cancel the common factor.

$4 (2 - \frac{2 \cdot 0}{2 \cdot 1}) - (\frac{2^{4}}{4} - \frac{0^{4}}{4})$

Step 5.9.2.3.4.2.3

Rewrite the expression.

$4 (2 - \frac{0}{1}) - (\frac{2^{4}}{4} - \frac{0^{4}}{4})$

Step 5.9.2.3.4.2.4

Divide $0$ by $1$ .

$4 (2 - 0) - (\frac{2^{4}}{4} - \frac{0^{4}}{4})$

Step 5.9.2.3.5

Multiply $- 1$ by $0$ .

$4 (2 + 0) - (\frac{2^{4}}{4} - \frac{0^{4}}{4})$

Step 5.9.2.3.6

Add $2$ and $0$ .

$4 \cdot 2 - (\frac{2^{4}}{4} - \frac{0^{4}}{4})$

Step 5.9.2.3.7

Multiply $4$ by $2$ .

$8 - (\frac{2^{4}}{4} - \frac{0^{4}}{4})$

Step 5.9.2.3.8

Raise $2$ to the power of $4$ .

$8 - (\frac{16}{4} - \frac{0^{4}}{4})$

Step 5.9.2.3.9

Cancel the common factor of $16$ and $4$ .

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Step 5.9.2.3.9.1

Factor $4$ out of $16$ .

$8 - (\frac{4 \cdot 4}{4} - \frac{0^{4}}{4})$

Step 5.9.2.3.9.2

Cancel the common factors.

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Step 5.9.2.3.9.2.1

Factor $4$ out of $4$ .

$8 - (\frac{4 \cdot 4}{4 (1)} - \frac{0^{4}}{4})$

Step 5.9.2.3.9.2.2

Cancel the common factor.

$8 - (\frac{4 \cdot 4}{4 \cdot 1} - \frac{0^{4}}{4})$

Step 5.9.2.3.9.2.3

Rewrite the expression.

$8 - (\frac{4}{1} - \frac{0^{4}}{4})$

Step 5.9.2.3.9.2.4

Divide $4$ by $1$ .

$8 - (4 - \frac{0^{4}}{4})$

Step 5.9.2.3.10

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

$8 - (4 - \frac{0}{4})$

Step 5.9.2.3.11

Cancel the common factor of $0$ and $4$ .

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Step 5.9.2.3.11.1

Factor $4$ out of $0$ .

$8 - (4 - \frac{4 (0)}{4})$

Step 5.9.2.3.11.2

Cancel the common factors.

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Step 5.9.2.3.11.2.1

Factor $4$ out of $4$ .

$8 - (4 - \frac{4 \cdot 0}{4 \cdot 1})$

Step 5.9.2.3.11.2.2

Cancel the common factor.

$8 - (4 - \frac{4 \cdot 0}{4 \cdot 1})$

Step 5.9.2.3.11.2.3

Rewrite the expression.

$8 - (4 - \frac{0}{1})$

Step 5.9.2.3.11.2.4

Divide $0$ by $1$ .

$8 - (4 - 0)$

Step 5.9.2.3.12

Multiply $- 1$ by $0$ .

$8 - (4 + 0)$

Step 5.9.2.3.13

Add $4$ and $0$ .

$8 - 1 \cdot 4$

Step 5.9.2.3.14

Multiply $- 1$ by $4$ .

$8 - 4$

Step 5.9.2.3.15

Subtract $4$ from $8$ .

$4$

Step 6

Add $4$ and $4$ .

$8$

Step 7