Calculus Examples

$y = 81 x$ , $y = x^{5}$

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.

$81 x = x^{5}$

Step 1.2

Solve $81 x = x^{5}$ for $x$ .

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Step 1.2.1

Subtract $x^{5}$ from both sides of the equation.

$81 x - x^{5} = 0$

Step 1.2.2

Factor the left side of the equation.

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Step 1.2.2.1

Factor $x$ out of $81 x - x^{5}$ .

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Step 1.2.2.1.1

Factor $x$ out of $81 x$ .

$x \cdot 81 - x^{5} = 0$

Step 1.2.2.1.2

Factor $x$ out of $- x^{5}$ .

$x \cdot 81 + x (- x^{4}) = 0$

Step 1.2.2.1.3

Factor $x$ out of $x \cdot 81 + x (- x^{4})$ .

$x (81 - x^{4}) = 0$

Step 1.2.2.2

Rewrite $81$ as $9^{2}$ .

$x (9^{2} - x^{4}) = 0$

Step 1.2.2.3

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$x (9^{2} - {(x^{2})}^{2}) = 0$

Step 1.2.2.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 9$ and $b = x^{2}$ .

$x ((9 + x^{2}) (9 - x^{2})) = 0$

Step 1.2.2.5

Factor.

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Step 1.2.2.5.1

Simplify.

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Step 1.2.2.5.1.1

Rewrite $9$ as $3^{2}$ .

$x ((9 + x^{2}) (3^{2} - x^{2})) = 0$

Step 1.2.2.5.1.2

Factor.

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Step 1.2.2.5.1.2.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3$ and $b = x$ .

$x ((9 + x^{2}) ((3 + x) (3 - x))) = 0$

Step 1.2.2.5.1.2.2

Remove unnecessary parentheses.

$x ((9 + x^{2}) (3 + x) (3 - x)) = 0$

Step 1.2.2.5.2

Remove unnecessary parentheses.

$x (9 + x^{2}) (3 + x) (3 - x) = 0$

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 $0$ .

$x = 0$

$9 + x^{2} = 0$

$3 + x = 0$

$3 - x = 0 + y = x^{5}$

Step 1.2.4

Set $x$ equal to $0$ .

$x = 0$

Step 1.2.5

Set $9 + x^{2}$ equal to $0$ and solve for $x$ .

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Step 1.2.5.1

Set $9 + x^{2}$ equal to $0$ .

$9 + x^{2} = 0$

Step 1.2.5.2

Solve $9 + x^{2} = 0$ for $x$ .

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Step 1.2.5.2.1

Subtract $9$ from both sides of the equation.

$x^{2} = - 9$

Step 1.2.5.2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{- 9}$

Step 1.2.5.2.3

Simplify $\pm \sqrt{- 9}$ .

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Step 1.2.5.2.3.1

Rewrite $- 9$ as $- 1 (9)$ .

$x = \pm \sqrt{- 1 (9)}$

Step 1.2.5.2.3.2

Rewrite $\sqrt{- 1 (9)}$ as $\sqrt{- 1} \cdot \sqrt{9}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{9}$

Step 1.2.5.2.3.3

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i \cdot \sqrt{9}$

Step 1.2.5.2.3.4

Rewrite $9$ as $3^{2}$ .

$x = \pm i \cdot \sqrt{3^{2}}$

Step 1.2.5.2.3.5

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm i \cdot 3$

Step 1.2.5.2.3.6

Move $3$ to the left of $i$ .

$x = \pm 3 i$

Step 1.2.5.2.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.2.5.2.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 3 i$

Step 1.2.5.2.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 3 i$

Step 1.2.5.2.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 3 i, - 3 i$

Step 1.2.6

Set $3 + x$ equal to $0$ and solve for $x$ .

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Step 1.2.6.1

Set $3 + x$ equal to $0$ .

$3 + x = 0$

Step 1.2.6.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 1.2.7

Set $3 - x$ equal to $0$ and solve for $x$ .

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Step 1.2.7.1

Set $3 - x$ equal to $0$ .

$3 - x = 0$

Step 1.2.7.2

Solve $3 - x = 0$ for $x$ .

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Step 1.2.7.2.1

Subtract $3$ from both sides of the equation.

$- x = - 3$

Step 1.2.7.2.2

Divide each term in $- x = - 3$ by $- 1$ and simplify.

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Step 1.2.7.2.2.1

Divide each term in $- x = - 3$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 3}{- 1}$

Step 1.2.7.2.2.2

Simplify the left side.

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Step 1.2.7.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 3}{- 1}$

Step 1.2.7.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 3}{- 1}$

Step 1.2.7.2.2.3

Simplify the right side.

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Step 1.2.7.2.2.3.1

Divide $- 3$ by $- 1$ .

$x = 3$

Step 1.2.8

The final solution is all the values that make $x (9 + x^{2}) (3 + x) (3 - x) = 0$ true.

$x = 0, 3 i, - 3 i, - 3, 3$

Step 1.3

Evaluate $y$ when $x = 0$ .

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Step 1.3.1

Substitute $0$ for $x$ .

$y = {(0)}^{5}$

Step 1.3.2

Substitute $0$ for $x$ in $y = {(0)}^{5}$ and solve for $y$ .

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Step 1.3.2.1

Remove parentheses.

$y = 0^{5}$

Step 1.3.2.2

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

$y = 0$

Step 1.4

Evaluate $y$ when $x = 3 i$ .

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Step 1.4.1

Substitute $3 i$ for $x$ .

$y = {(3 i)}^{5}$

Step 1.4.2

Simplify ${(3 i)}^{5}$ .

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Step 1.4.2.1

Apply the product rule to $3 i$ .

$y = 3^{5} i^{5}$

Step 1.4.2.2

Raise $3$ to the power of $5$ .

$y = 243 i^{5}$

Step 1.4.2.3

Factor out $i^{4}$ .

$y = 243 (i^{4} i)$

Step 1.4.2.4

Rewrite $i^{4}$ as $1$ .

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Step 1.4.2.4.1

Rewrite $i^{4}$ as ${(i^{2})}^{2}$ .

$y = 243 ({(i^{2})}^{2} i)$

Step 1.4.2.4.2

Rewrite $i^{2}$ as $- 1$ .

$y = 243 ({(- 1)}^{2} i)$

Step 1.4.2.4.3

Raise $- 1$ to the power of $2$ .

$y = 243 (1 i)$

Step 1.4.2.5

Multiply $i$ by $1$ .

$y = 243 i$

Step 1.5

Evaluate $y$ when $x = - 3 i$ .

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Step 1.5.1

Substitute $- 3 i$ for $x$ .

$y = {(- 3 i)}^{5}$

Step 1.5.2

Simplify ${(- 3 i)}^{5}$ .

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Step 1.5.2.1

Apply the product rule to $- 3 i$ .

$y = {(- 3)}^{5} i^{5}$

Step 1.5.2.2

Raise $- 3$ to the power of $5$ .

$y = - 243 i^{5}$

Step 1.5.2.3

Factor out $i^{4}$ .

$y = - 243 (i^{4} i)$

Step 1.5.2.4

Rewrite $i^{4}$ as $1$ .

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Step 1.5.2.4.1

Rewrite $i^{4}$ as ${(i^{2})}^{2}$ .

$y = - 243 ({(i^{2})}^{2} i)$

Step 1.5.2.4.2

Rewrite $i^{2}$ as $- 1$ .

$y = - 243 ({(- 1)}^{2} i)$

Step 1.5.2.4.3

Raise $- 1$ to the power of $2$ .

$y = - 243 (1 i)$

Step 1.5.2.5

Multiply $i$ by $1$ .

$y = - 243 i$

Step 1.6

Evaluate $y$ when $x = - 3$ .

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Step 1.6.1

Substitute $- 3$ for $x$ .

$y = {(- 3)}^{5}$

Step 1.6.2

Raise $- 3$ to the power of $5$ .

$y = - 243$

Step 1.7

Evaluate $y$ when $x = 3$ .

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Step 1.7.1

Substitute $3$ for $x$ .

$y = {(3)}^{5}$

Step 1.7.2

Substitute $3$ for $x$ in $y = {(3)}^{5}$ and solve for $y$ .

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Step 1.7.2.1

Remove parentheses.

$y = 3^{5}$

Step 1.7.2.2

Raise $3$ to the power of $5$ .

$y = 243$

Step 1.8

List all of the solutions.

$y = 0, x = 0$

$y = 243 i, x = 3 i$

$y = - 243 i, x = - 3 i$

$y = - 243, x = - 3$

$y = 243, x = 3$

$y = 0, x = 0$

$y = 243 i, x = 3 i$

$y = - 243 i, x = - 3 i$

$y = - 243, x = - 3$

$y = 243, x = 3$

Step 2

The area between the given curves is unbounded.

Unbounded area

Step 3