Calculus Examples

$y = 81 x$ , $y = x^{5}$ , $x = 0$ , $x = 3$

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 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.

$A r e a = \int_{0}^{3} 81 x d x - \int_{0}^{3} x^{5} d x$

Step 3

Integrate to find the area between $0$ and $3$ .

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Step 3.1

Combine the integrals into a single integral.

$\int_{0}^{3} 81 x - (x^{5}) d x$

Step 3.2

Multiply $- 1$ by $x^{5}$ .

$\int_{0}^{3} 81 x - x^{5} d x$

Step 3.3

Split the single integral into multiple integrals.

$\int_{0}^{3} 81 x d x + \int_{0}^{3} - x^{5} d x$

Step 3.4

Since $81$ is constant with respect to $x$ , move $81$ out of the integral.

$81 \int_{0}^{3} x d x + \int_{0}^{3} - x^{5} d x$

Step 3.5

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

$81 (\frac{1}{2} x^{2}]_{0}^{3}) + \int_{0}^{3} - x^{5} d x$

Step 3.6

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

$81 (\frac{x^{2}}{2}]_{0}^{3}) + \int_{0}^{3} - x^{5} d x$

Step 3.7

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

$81 (\frac{x^{2}}{2}]_{0}^{3}) - \int_{0}^{3} x^{5} d x$

Step 3.8

By the Power Rule, the integral of $x^{5}$ with respect to $x$ is $\frac{1}{6} x^{6}$ .

$81 (\frac{x^{2}}{2}]_{0}^{3}) - (\frac{1}{6} x^{6}]_{0}^{3})$

Step 3.9

Simplify the answer.

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Step 3.9.1

Combine $\frac{1}{6}$ and $x^{6}$ .

$81 (\frac{x^{2}}{2}]_{0}^{3}) - (\frac{x^{6}}{6}]_{0}^{3})$

Step 3.9.2

Substitute and simplify.

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Step 3.9.2.1

Evaluate $\frac{x^{2}}{2}$ at $3$ and at $0$ .

$81 ((\frac{3^{2}}{2}) - \frac{0^{2}}{2}) - (\frac{x^{6}}{6}]_{0}^{3})$

Step 3.9.2.2

Evaluate $\frac{x^{6}}{6}$ at $3$ and at $0$ .

$81 (\frac{3^{2}}{2} - \frac{0^{2}}{2}) - (\frac{3^{6}}{6} - \frac{0^{6}}{6})$

Step 3.9.2.3

Simplify.

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Step 3.9.2.3.1

Raise $3$ to the power of $2$ .

$81 (\frac{9}{2} - \frac{0^{2}}{2}) - (\frac{3^{6}}{6} - \frac{0^{6}}{6})$

Step 3.9.2.3.2

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

$81 (\frac{9}{2} - \frac{0}{2}) - (\frac{3^{6}}{6} - \frac{0^{6}}{6})$

Step 3.9.2.3.3

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

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Step 3.9.2.3.3.1

Factor $2$ out of $0$ .

$81 (\frac{9}{2} - \frac{2 (0)}{2}) - (\frac{3^{6}}{6} - \frac{0^{6}}{6})$

Step 3.9.2.3.3.2

Cancel the common factors.

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Step 3.9.2.3.3.2.1

Factor $2$ out of $2$ .

$81 (\frac{9}{2} - \frac{2 \cdot 0}{2 \cdot 1}) - (\frac{3^{6}}{6} - \frac{0^{6}}{6})$

Step 3.9.2.3.3.2.2

Cancel the common factor.

$81 (\frac{9}{2} - \frac{2 \cdot 0}{2 \cdot 1}) - (\frac{3^{6}}{6} - \frac{0^{6}}{6})$

Step 3.9.2.3.3.2.3

Rewrite the expression.

$81 (\frac{9}{2} - \frac{0}{1}) - (\frac{3^{6}}{6} - \frac{0^{6}}{6})$

Step 3.9.2.3.3.2.4

Divide $0$ by $1$ .

$81 (\frac{9}{2} - 0) - (\frac{3^{6}}{6} - \frac{0^{6}}{6})$

Step 3.9.2.3.4

Multiply $- 1$ by $0$ .

$81 (\frac{9}{2} + 0) - (\frac{3^{6}}{6} - \frac{0^{6}}{6})$

Step 3.9.2.3.5

Add $\frac{9}{2}$ and $0$ .

$81 (\frac{9}{2}) - (\frac{3^{6}}{6} - \frac{0^{6}}{6})$

Step 3.9.2.3.6

Combine $81$ and $\frac{9}{2}$ .

$\frac{81 \cdot 9}{2} - (\frac{3^{6}}{6} - \frac{0^{6}}{6})$

Step 3.9.2.3.7

Multiply $81$ by $9$ .

$\frac{729}{2} - (\frac{3^{6}}{6} - \frac{0^{6}}{6})$

Step 3.9.2.3.8

Raise $3$ to the power of $6$ .

$\frac{729}{2} - (\frac{729}{6} - \frac{0^{6}}{6})$

Step 3.9.2.3.9

Cancel the common factor of $729$ and $6$ .

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Step 3.9.2.3.9.1

Factor $3$ out of $729$ .

$\frac{729}{2} - (\frac{3 (243)}{6} - \frac{0^{6}}{6})$

Step 3.9.2.3.9.2

Cancel the common factors.

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Step 3.9.2.3.9.2.1

Factor $3$ out of $6$ .

$\frac{729}{2} - (\frac{3 \cdot 243}{3 \cdot 2} - \frac{0^{6}}{6})$

Step 3.9.2.3.9.2.2

Cancel the common factor.

$\frac{729}{2} - (\frac{3 \cdot 243}{3 \cdot 2} - \frac{0^{6}}{6})$

Step 3.9.2.3.9.2.3

Rewrite the expression.

$\frac{729}{2} - (\frac{243}{2} - \frac{0^{6}}{6})$

Step 3.9.2.3.10

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

$\frac{729}{2} - (\frac{243}{2} - \frac{0}{6})$

Step 3.9.2.3.11

Cancel the common factor of $0$ and $6$ .

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Step 3.9.2.3.11.1

Factor $6$ out of $0$ .

$\frac{729}{2} - (\frac{243}{2} - \frac{6 (0)}{6})$

Step 3.9.2.3.11.2

Cancel the common factors.

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Step 3.9.2.3.11.2.1

Factor $6$ out of $6$ .

$\frac{729}{2} - (\frac{243}{2} - \frac{6 \cdot 0}{6 \cdot 1})$

Step 3.9.2.3.11.2.2

Cancel the common factor.

$\frac{729}{2} - (\frac{243}{2} - \frac{6 \cdot 0}{6 \cdot 1})$

Step 3.9.2.3.11.2.3

Rewrite the expression.

$\frac{729}{2} - (\frac{243}{2} - \frac{0}{1})$

Step 3.9.2.3.11.2.4

Divide $0$ by $1$ .

$\frac{729}{2} - (\frac{243}{2} - 0)$

Step 3.9.2.3.12

Multiply $- 1$ by $0$ .

$\frac{729}{2} - (\frac{243}{2} + 0)$

Step 3.9.2.3.13

Add $\frac{243}{2}$ and $0$ .

$\frac{729}{2} - \frac{243}{2}$

Step 3.9.2.3.14

Combine the numerators over the common denominator.

$\frac{729 - 243}{2}$

Step 3.9.2.3.15

Subtract $243$ from $729$ .

$\frac{486}{2}$

Step 3.9.2.3.16

Cancel the common factor of $486$ and $2$ .

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Step 3.9.2.3.16.1

Factor $2$ out of $486$ .

$\frac{2 \cdot 243}{2}$

Step 3.9.2.3.16.2

Cancel the common factors.

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Step 3.9.2.3.16.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 243}{2 (1)}$

Step 3.9.2.3.16.2.2

Cancel the common factor.

$\frac{2 \cdot 243}{2 \cdot 1}$

Step 3.9.2.3.16.2.3

Rewrite the expression.

$\frac{243}{1}$

Step 3.9.2.3.16.2.4

Divide $243$ by $1$ .

$243$

Step 4