Calculus Examples

$y = x - \frac{1}{75} x^{3}$ , $y = 0$

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.

$x - \frac{1}{75} x^{3} = 0$

Step 1.2

Solve $x - \frac{1}{75} x^{3} = 0$ for $x$ .

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Step 1.2.1

Combine $x^{3}$ and $\frac{1}{75}$ .

$x - \frac{x^{3}}{75} = 0$

Step 1.2.2

Factor $x$ out of $x - \frac{x^{3}}{75}$ .

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Step 1.2.2.1

Factor $x$ out of $x^{1}$ .

$x \cdot 1 - \frac{x^{3}}{75} = 0$

Step 1.2.2.2

Factor $x$ out of $- \frac{x^{3}}{75}$ .

$x \cdot 1 + x (- \frac{x^{2}}{75}) = 0$

Step 1.2.2.3

Factor $x$ out of $x \cdot 1 + x (- \frac{x^{2}}{75})$ .

$x (1 - \frac{x^{2}}{75}) = 0$

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 $0$ .

$x = 0$

$1 - \frac{x^{2}}{75} = 0 + y = 0$

Step 1.2.4

Set $x$ equal to $0$ .

$x = 0$

Step 1.2.5

Set $1 - \frac{x^{2}}{75}$ equal to $0$ and solve for $x$ .

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Step 1.2.5.1

Set $1 - \frac{x^{2}}{75}$ equal to $0$ .

$1 - \frac{x^{2}}{75} = 0$

Step 1.2.5.2

Solve $1 - \frac{x^{2}}{75} = 0$ for $x$ .

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Step 1.2.5.2.1

Subtract $1$ from both sides of the equation.

$- \frac{x^{2}}{75} = - 1$

Step 1.2.5.2.2

Multiply both sides of the equation by $- 75$ .

$- 75 (- \frac{x^{2}}{75}) = - 75 \cdot - 1$

Step 1.2.5.2.3

Simplify both sides of the equation.

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Step 1.2.5.2.3.1

Simplify the left side.

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Step 1.2.5.2.3.1.1

Simplify $- 75 (- \frac{x^{2}}{75})$ .

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Step 1.2.5.2.3.1.1.1

Cancel the common factor of $75$ .

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Step 1.2.5.2.3.1.1.1.1

Move the leading negative in $- \frac{x^{2}}{75}$ into the numerator.

$- 75 \frac{- x^{2}}{75} = - 75 \cdot - 1$

Step 1.2.5.2.3.1.1.1.2

Factor $75$ out of $- 75$ .

$75 (- 1) \frac{- x^{2}}{75} = - 75 \cdot - 1$

Step 1.2.5.2.3.1.1.1.3

Cancel the common factor.

$75 \cdot - 1 \frac{- x^{2}}{75} = - 75 \cdot - 1$

Step 1.2.5.2.3.1.1.1.4

Rewrite the expression.

$- - x^{2} = - 75 \cdot - 1$

Step 1.2.5.2.3.1.1.2

Multiply.

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Step 1.2.5.2.3.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 x^{2} = - 75 \cdot - 1$

Step 1.2.5.2.3.1.1.2.2

Multiply $x^{2}$ by $1$ .

$x^{2} = - 75 \cdot - 1$

Step 1.2.5.2.3.2

Simplify the right side.

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Step 1.2.5.2.3.2.1

Multiply $- 75$ by $- 1$ .

$x^{2} = 75$

Step 1.2.5.2.4

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

$x = \pm \sqrt{75}$

Step 1.2.5.2.5

Simplify $\pm \sqrt{75}$ .

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Step 1.2.5.2.5.1

Rewrite $75$ as $5^{2} \cdot 3$ .

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Step 1.2.5.2.5.1.1

Factor $25$ out of $75$ .

$x = \pm \sqrt{25 (3)}$

Step 1.2.5.2.5.1.2

Rewrite $25$ as $5^{2}$ .

$x = \pm \sqrt{5^{2} \cdot 3}$

Step 1.2.5.2.5.2

Pull terms out from under the radical.

$x = \pm 5 \sqrt{3}$

Step 1.2.5.2.6

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

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Step 1.2.5.2.6.1

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

$x = 5 \sqrt{3}$

Step 1.2.5.2.6.2

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

$x = - 5 \sqrt{3}$

Step 1.2.5.2.6.3

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

$x = 5 \sqrt{3}, - 5 \sqrt{3}$

Step 1.2.6

The final solution is all the values that make $x (1 - \frac{x^{2}}{75}) = 0$ true.

$x = 0, 5 \sqrt{3}, - 5 \sqrt{3}$

Step 1.3

Substitute $0$ for $x$ .

$y = 0$

Step 1.4

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

$(0, 0)$

$(5 \sqrt{3}, 0)$

$(- 5 \sqrt{3}, 0)$

$(0, 0)$

$(5 \sqrt{3}, 0)$

$(- 5 \sqrt{3}, 0)$

Step 2

Simplify $x - \frac{1}{75} x^{3}$ .

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Step 2.1

Combine $x^{3}$ and $\frac{1}{75}$ .

$y = x - \frac{x^{3}}{75}$

Step 2.2

Reorder $x$ and $- \frac{x^{3}}{75}$ .

$y = - \frac{x^{3}}{75} + x$

Step 3

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.

$A r e a = \int_{- 5 \sqrt{3}}^{0} 0 d x - \int_{- 5 \sqrt{3}}^{0} - \frac{x^{3}}{75} + x d x$

Step 4

Integrate to find the area between $- 5 \sqrt{3}$ and $0$ .

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Step 4.1

Combine the integrals into a single integral.

$\int_{- 5 \sqrt{3}}^{0} 0 - (- \frac{x^{3}}{75} + x) d x$

Step 4.2

Subtract $- \frac{x^{3}}{75} + x$ from $0$ .

$\int_{- 5 \sqrt{3}}^{0} - (- \frac{x^{3}}{75} + x) d x$

Step 4.3

Apply the distributive property.

$\int_{- 5 \sqrt{3}}^{0} \frac{x^{3}}{75} - x d x$

Step 4.4

Split the single integral into multiple integrals.

$\int_{- 5 \sqrt{3}}^{0} \frac{x^{3}}{75} d x + \int_{- 5 \sqrt{3}}^{0} - x d x$

Step 4.5

Since $\frac{1}{75}$ is constant with respect to $x$ , move $\frac{1}{75}$ out of the integral.

$\frac{1}{75} \int_{- 5 \sqrt{3}}^{0} x^{3} d x + \int_{- 5 \sqrt{3}}^{0} - x d x$

Step 4.6

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

$\frac{1}{75} \frac{1}{4} x^{4}]_{- 5 \sqrt{3}}^{0} + \int_{- 5 \sqrt{3}}^{0} - x d x$

Step 4.7

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

$\frac{1}{75} \frac{1}{4} x^{4}]_{- 5 \sqrt{3}}^{0} - \int_{- 5 \sqrt{3}}^{0} x d x$

Step 4.8

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

$\frac{1}{75} \frac{1}{4} x^{4}]_{- 5 \sqrt{3}}^{0} - (\frac{1}{2} x^{2}]_{- 5 \sqrt{3}}^{0})$

Step 4.9

Simplify the answer.

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Step 4.9.1

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

$\frac{1}{75} \frac{1}{4} x^{4}]_{- 5 \sqrt{3}}^{0} - (\frac{x^{2}}{2}]_{- 5 \sqrt{3}}^{0})$

Step 4.9.2

Substitute and simplify.

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Step 4.9.2.1

Evaluate $\frac{1}{4} x^{4}$ at $0$ and at $- 5 \sqrt{3}$ .

$\frac{1}{75} ((\frac{1}{4} \cdot 0^{4}) - \frac{1}{4} {(- 5 \sqrt{3})}^{4}) - (\frac{x^{2}}{2}]_{- 5 \sqrt{3}}^{0})$

Step 4.9.2.2

Evaluate $\frac{x^{2}}{2}$ at $0$ and at $- 5 \sqrt{3}$ .

$\frac{1}{75} (\frac{1}{4} \cdot 0^{4} - \frac{1}{4} {(- 5 \sqrt{3})}^{4}) - (\frac{0^{2}}{2} - \frac{{(- 5 \sqrt{3})}^{2}}{2})$

Step 4.9.2.3

Simplify.

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Step 4.9.2.3.1

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

$\frac{1}{75} (\frac{1}{4} \cdot 0 - \frac{1}{4} {(- 5 \sqrt{3})}^{4}) - (\frac{0^{2}}{2} - \frac{{(- 5 \sqrt{3})}^{2}}{2})$

Step 4.9.2.3.2

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

$\frac{1}{75} (0 - \frac{1}{4} {(- 5 \sqrt{3})}^{4}) - (\frac{0^{2}}{2} - \frac{{(- 5 \sqrt{3})}^{2}}{2})$

Step 4.9.2.3.3

Factor $- 5$ out of $- 5 \sqrt{3}$ .

$\frac{1}{75} (0 - \frac{1}{4} {(- 5 (\sqrt{3}))}^{4}) - (\frac{0^{2}}{2} - \frac{{(- 5 \sqrt{3})}^{2}}{2})$

Step 4.9.2.3.4

Apply the product rule to $- 5 (\sqrt{3})$ .

$\frac{1}{75} (0 - \frac{1}{4} ({(- 5)}^{4} {\sqrt{3}}^{4})) - (\frac{0^{2}}{2} - \frac{{(- 5 \sqrt{3})}^{2}}{2})$

Step 4.9.2.3.5

Raise $- 5$ to the power of $4$ .

$\frac{1}{75} (0 - \frac{1}{4} (625 {\sqrt{3}}^{4})) - (\frac{0^{2}}{2} - \frac{{(- 5 \sqrt{3})}^{2}}{2})$

Step 4.9.2.3.6

Rewrite ${\sqrt{3}}^{4}$ as $3^{2}$ .

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Step 4.9.2.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{1}{75} (0 - \frac{1}{4} (625 {(3^{\frac{1}{2}})}^{4})) - (\frac{0^{2}}{2} - \frac{{(- 5 \sqrt{3})}^{2}}{2})$

Step 4.9.2.3.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{75} (0 - \frac{1}{4} (625 \cdot 3^{\frac{1}{2} \cdot 4})) - (\frac{0^{2}}{2} - \frac{{(- 5 \sqrt{3})}^{2}}{2})$

Step 4.9.2.3.6.3

Combine $\frac{1}{2}$ and $4$ .

$\frac{1}{75} (0 - \frac{1}{4} (625 \cdot 3^{\frac{4}{2}})) - (\frac{0^{2}}{2} - \frac{{(- 5 \sqrt{3})}^{2}}{2})$

Step 4.9.2.3.6.4

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

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Step 4.9.2.3.6.4.1

Factor $2$ out of $4$ .

$\frac{1}{75} (0 - \frac{1}{4} (625 \cdot 3^{\frac{2 \cdot 2}{2}})) - (\frac{0^{2}}{2} - \frac{{(- 5 \sqrt{3})}^{2}}{2})$

Step 4.9.2.3.6.4.2

Cancel the common factors.

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Step 4.9.2.3.6.4.2.1

Factor $2$ out of $2$ .

$\frac{1}{75} (0 - \frac{1}{4} (625 \cdot 3^{\frac{2 \cdot 2}{2 (1)}})) - (\frac{0^{2}}{2} - \frac{{(- 5 \sqrt{3})}^{2}}{2})$

Step 4.9.2.3.6.4.2.2

Cancel the common factor.

$\frac{1}{75} (0 - \frac{1}{4} (625 \cdot 3^{\frac{2 \cdot 2}{2 \cdot 1}})) - (\frac{0^{2}}{2} - \frac{{(- 5 \sqrt{3})}^{2}}{2})$

Step 4.9.2.3.6.4.2.3

Rewrite the expression.

$\frac{1}{75} (0 - \frac{1}{4} (625 \cdot 3^{\frac{2}{1}})) - (\frac{0^{2}}{2} - \frac{{(- 5 \sqrt{3})}^{2}}{2})$

Step 4.9.2.3.6.4.2.4

Divide $2$ by $1$ .

$\frac{1}{75} (0 - \frac{1}{4} (625 \cdot 3^{2})) - (\frac{0^{2}}{2} - \frac{{(- 5 \sqrt{3})}^{2}}{2})$

Step 4.9.2.3.7

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

$\frac{1}{75} (0 - \frac{1}{4} (625 \cdot 9)) - (\frac{0^{2}}{2} - \frac{{(- 5 \sqrt{3})}^{2}}{2})$

Step 4.9.2.3.8

Multiply $625$ by $9$ .

$\frac{1}{75} (0 - \frac{1}{4} \cdot 5625) - (\frac{0^{2}}{2} - \frac{{(- 5 \sqrt{3})}^{2}}{2})$

Step 4.9.2.3.9

Multiply $5625$ by $- 1$ .

$\frac{1}{75} (0 - 5625 (\frac{1}{4})) - (\frac{0^{2}}{2} - \frac{{(- 5 \sqrt{3})}^{2}}{2})$

Step 4.9.2.3.10

Combine $- 5625$ and $\frac{1}{4}$ .

$\frac{1}{75} (0 + \frac{- 5625}{4}) - (\frac{0^{2}}{2} - \frac{{(- 5 \sqrt{3})}^{2}}{2})$

Step 4.9.2.3.11

Move the negative in front of the fraction.

$\frac{1}{75} (0 - \frac{5625}{4}) - (\frac{0^{2}}{2} - \frac{{(- 5 \sqrt{3})}^{2}}{2})$

Step 4.9.2.3.12

Subtract $\frac{5625}{4}$ from $0$ .

$\frac{1}{75} (- \frac{5625}{4}) - (\frac{0^{2}}{2} - \frac{{(- 5 \sqrt{3})}^{2}}{2})$

Step 4.9.2.3.13

Multiply $\frac{1}{75}$ by $\frac{5625}{4}$ .

$- \frac{5625}{75 \cdot 4} - (\frac{0^{2}}{2} - \frac{{(- 5 \sqrt{3})}^{2}}{2})$

Step 4.9.2.3.14

Multiply $75$ by $4$ .

$- \frac{5625}{300} - (\frac{0^{2}}{2} - \frac{{(- 5 \sqrt{3})}^{2}}{2})$

Step 4.9.2.3.15

Cancel the common factor of $5625$ and $300$ .

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Step 4.9.2.3.15.1

Factor $75$ out of $5625$ .

$- \frac{75 (75)}{300} - (\frac{0^{2}}{2} - \frac{{(- 5 \sqrt{3})}^{2}}{2})$

Step 4.9.2.3.15.2

Cancel the common factors.

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Step 4.9.2.3.15.2.1

Factor $75$ out of $300$ .

$- \frac{75 \cdot 75}{75 \cdot 4} - (\frac{0^{2}}{2} - \frac{{(- 5 \sqrt{3})}^{2}}{2})$

Step 4.9.2.3.15.2.2

Cancel the common factor.

$- \frac{75 \cdot 75}{75 \cdot 4} - (\frac{0^{2}}{2} - \frac{{(- 5 \sqrt{3})}^{2}}{2})$

Step 4.9.2.3.15.2.3

Rewrite the expression.

$- \frac{75}{4} - (\frac{0^{2}}{2} - \frac{{(- 5 \sqrt{3})}^{2}}{2})$

Step 4.9.2.3.16

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

$- \frac{75}{4} - (\frac{0}{2} - \frac{{(- 5 \sqrt{3})}^{2}}{2})$

Step 4.9.2.3.17

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

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Step 4.9.2.3.17.1

Factor $2$ out of $0$ .

$- \frac{75}{4} - (\frac{2 (0)}{2} - \frac{{(- 5 \sqrt{3})}^{2}}{2})$

Step 4.9.2.3.17.2

Cancel the common factors.

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Step 4.9.2.3.17.2.1

Factor $2$ out of $2$ .

$- \frac{75}{4} - (\frac{2 \cdot 0}{2 \cdot 1} - \frac{{(- 5 \sqrt{3})}^{2}}{2})$

Step 4.9.2.3.17.2.2

Cancel the common factor.

$- \frac{75}{4} - (\frac{2 \cdot 0}{2 \cdot 1} - \frac{{(- 5 \sqrt{3})}^{2}}{2})$

Step 4.9.2.3.17.2.3

Rewrite the expression.

$- \frac{75}{4} - (\frac{0}{1} - \frac{{(- 5 \sqrt{3})}^{2}}{2})$

Step 4.9.2.3.17.2.4

Divide $0$ by $1$ .

$- \frac{75}{4} - (0 - \frac{{(- 5 \sqrt{3})}^{2}}{2})$

Step 4.9.2.3.18

Factor $- 5$ out of $- 5 \sqrt{3}$ .

$- \frac{75}{4} - (0 - \frac{{(- 5 (\sqrt{3}))}^{2}}{2})$

Step 4.9.2.3.19

Apply the product rule to $- 5 (\sqrt{3})$ .

$- \frac{75}{4} - (0 - \frac{{(- 5)}^{2} {\sqrt{3}}^{2}}{2})$

Step 4.9.2.3.20

Raise $- 5$ to the power of $2$ .

$- \frac{75}{4} - (0 - \frac{25 {\sqrt{3}}^{2}}{2})$

Step 4.9.2.3.21

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 4.9.2.3.21.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$- \frac{75}{4} - (0 - \frac{25 {(3^{\frac{1}{2}})}^{2}}{2})$

Step 4.9.2.3.21.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- \frac{75}{4} - (0 - \frac{25 \cdot 3^{\frac{1}{2} \cdot 2}}{2})$

Step 4.9.2.3.21.3

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

$- \frac{75}{4} - (0 - \frac{25 \cdot 3^{\frac{2}{2}}}{2})$

Step 4.9.2.3.21.4

Cancel the common factor of $2$ .

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Step 4.9.2.3.21.4.1

Cancel the common factor.

$- \frac{75}{4} - (0 - \frac{25 \cdot 3^{\frac{2}{2}}}{2})$

Step 4.9.2.3.21.4.2

Rewrite the expression.

$- \frac{75}{4} - (0 - \frac{25 \cdot 3^{1}}{2})$

Step 4.9.2.3.21.5

Evaluate the exponent.

$- \frac{75}{4} - (0 - \frac{25 \cdot 3}{2})$

Step 4.9.2.3.22

Multiply $25$ by $3$ .

$- \frac{75}{4} - (0 - \frac{75}{2})$

Step 4.9.2.3.23

Subtract $\frac{75}{2}$ from $0$ .

$- \frac{75}{4} - - \frac{75}{2}$

Step 4.9.2.3.24

Multiply $- 1$ by $- 1$ .

$- \frac{75}{4} + 1 (\frac{75}{2})$

Step 4.9.2.3.25

Multiply $\frac{75}{2}$ by $1$ .

$- \frac{75}{4} + \frac{75}{2}$

Step 4.9.2.3.26

To write $\frac{75}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{75}{4} + \frac{75}{2} \cdot \frac{2}{2}$

Step 4.9.2.3.27

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.9.2.3.27.1

Multiply $\frac{75}{2}$ by $\frac{2}{2}$ .

$- \frac{75}{4} + \frac{75 \cdot 2}{2 \cdot 2}$

Step 4.9.2.3.27.2

Multiply $2$ by $2$ .

$- \frac{75}{4} + \frac{75 \cdot 2}{4}$

Step 4.9.2.3.28

Combine the numerators over the common denominator.

$\frac{- 75 + 75 \cdot 2}{4}$

Step 4.9.2.3.29

Simplify the numerator.

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Step 4.9.2.3.29.1

Multiply $75$ by $2$ .

$\frac{- 75 + 150}{4}$

Step 4.9.2.3.29.2

Add $- 75$ and $150$ .

$\frac{75}{4}$

Step 5

$A r e a = \int_{0}^{5 \sqrt{3}} - \frac{x^{3}}{75} + x d x - \int_{0}^{5 \sqrt{3}} 0 d x$

Step 6

Integrate to find the area between $0$ and $5 \sqrt{3}$ .

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Step 6.1

Combine the integrals into a single integral.

$\int_{0}^{5 \sqrt{3}} - \frac{x^{3}}{75} + x - (0) d x$

Step 6.2

Subtract $0$ from $- \frac{x^{3}}{75} + x$ .

$\int_{0}^{5 \sqrt{3}} - \frac{x^{3}}{75} + x d x$

Step 6.3

Split the single integral into multiple integrals.

$\int_{0}^{5 \sqrt{3}} - \frac{x^{3}}{75} d x + \int_{0}^{5 \sqrt{3}} x d x$

Step 6.4

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

$- \int_{0}^{5 \sqrt{3}} \frac{x^{3}}{75} d x + \int_{0}^{5 \sqrt{3}} x d x$

Step 6.5

Since $\frac{1}{75}$ is constant with respect to $x$ , move $\frac{1}{75}$ out of the integral.

$- (\frac{1}{75} \int_{0}^{5 \sqrt{3}} x^{3} d x) + \int_{0}^{5 \sqrt{3}} x d x$

Step 6.6

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

$- \frac{1}{75} \frac{1}{4} x^{4}]_{0}^{5 \sqrt{3}} + \int_{0}^{5 \sqrt{3}} x d x$

Step 6.7

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

$- \frac{1}{75} \frac{1}{4} x^{4}]_{0}^{5 \sqrt{3}} + \frac{1}{2} x^{2}]_{0}^{5 \sqrt{3}}$

Step 6.8

Substitute and simplify.

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Step 6.8.1

Evaluate $\frac{1}{4} x^{4}$ at $5 \sqrt{3}$ and at $0$ .

$- \frac{1}{75} ((\frac{1}{4} {(5 \sqrt{3})}^{4}) - \frac{1}{4} \cdot 0^{4}) + \frac{1}{2} x^{2}]_{0}^{5 \sqrt{3}}$

Step 6.8.2

Evaluate $\frac{1}{2} x^{2}$ at $5 \sqrt{3}$ and at $0$ .

$- \frac{1}{75} (\frac{1}{4} {(5 \sqrt{3})}^{4} - \frac{1}{4} \cdot 0^{4}) + \frac{1}{2} {(5 \sqrt{3})}^{2} - \frac{1}{2} \cdot 0^{2}$

Step 6.8.3

Simplify.

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Step 6.8.3.1

Factor $5$ out of $5 \sqrt{3}$ .

$- \frac{1}{75} (\frac{1}{4} {(5 (\sqrt{3}))}^{4} - \frac{1}{4} \cdot 0^{4}) + \frac{1}{2} {(5 \sqrt{3})}^{2} - \frac{1}{2} \cdot 0^{2}$

Step 6.8.3.2

Apply the product rule to $5 (\sqrt{3})$ .

$- \frac{1}{75} (\frac{1}{4} (5^{4} {\sqrt{3}}^{4}) - \frac{1}{4} \cdot 0^{4}) + \frac{1}{2} {(5 \sqrt{3})}^{2} - \frac{1}{2} \cdot 0^{2}$

Step 6.8.3.3

Raise $5$ to the power of $4$ .

$- \frac{1}{75} (\frac{1}{4} (625 {\sqrt{3}}^{4}) - \frac{1}{4} \cdot 0^{4}) + \frac{1}{2} {(5 \sqrt{3})}^{2} - \frac{1}{2} \cdot 0^{2}$

Step 6.8.3.4

Rewrite ${\sqrt{3}}^{4}$ as $3^{2}$ .

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Step 6.8.3.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$- \frac{1}{75} (\frac{1}{4} (625 {(3^{\frac{1}{2}})}^{4}) - \frac{1}{4} \cdot 0^{4}) + \frac{1}{2} {(5 \sqrt{3})}^{2} - \frac{1}{2} \cdot 0^{2}$

Step 6.8.3.4.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- \frac{1}{75} (\frac{1}{4} (625 \cdot 3^{\frac{1}{2} \cdot 4}) - \frac{1}{4} \cdot 0^{4}) + \frac{1}{2} {(5 \sqrt{3})}^{2} - \frac{1}{2} \cdot 0^{2}$

Step 6.8.3.4.3

Combine $\frac{1}{2}$ and $4$ .

$- \frac{1}{75} (\frac{1}{4} (625 \cdot 3^{\frac{4}{2}}) - \frac{1}{4} \cdot 0^{4}) + \frac{1}{2} {(5 \sqrt{3})}^{2} - \frac{1}{2} \cdot 0^{2}$

Step 6.8.3.4.4

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

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Step 6.8.3.4.4.1

Factor $2$ out of $4$ .

$- \frac{1}{75} (\frac{1}{4} (625 \cdot 3^{\frac{2 \cdot 2}{2}}) - \frac{1}{4} \cdot 0^{4}) + \frac{1}{2} {(5 \sqrt{3})}^{2} - \frac{1}{2} \cdot 0^{2}$

Step 6.8.3.4.4.2

Cancel the common factors.

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Step 6.8.3.4.4.2.1

Factor $2$ out of $2$ .

$- \frac{1}{75} (\frac{1}{4} (625 \cdot 3^{\frac{2 \cdot 2}{2 (1)}}) - \frac{1}{4} \cdot 0^{4}) + \frac{1}{2} {(5 \sqrt{3})}^{2} - \frac{1}{2} \cdot 0^{2}$

Step 6.8.3.4.4.2.2

Cancel the common factor.

$- \frac{1}{75} (\frac{1}{4} (625 \cdot 3^{\frac{2 \cdot 2}{2 \cdot 1}}) - \frac{1}{4} \cdot 0^{4}) + \frac{1}{2} {(5 \sqrt{3})}^{2} - \frac{1}{2} \cdot 0^{2}$

Step 6.8.3.4.4.2.3

Rewrite the expression.

$- \frac{1}{75} (\frac{1}{4} (625 \cdot 3^{\frac{2}{1}}) - \frac{1}{4} \cdot 0^{4}) + \frac{1}{2} {(5 \sqrt{3})}^{2} - \frac{1}{2} \cdot 0^{2}$

Step 6.8.3.4.4.2.4

Divide $2$ by $1$ .

$- \frac{1}{75} (\frac{1}{4} (625 \cdot 3^{2}) - \frac{1}{4} \cdot 0^{4}) + \frac{1}{2} {(5 \sqrt{3})}^{2} - \frac{1}{2} \cdot 0^{2}$

Step 6.8.3.5

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

$- \frac{1}{75} (\frac{1}{4} (625 \cdot 9) - \frac{1}{4} \cdot 0^{4}) + \frac{1}{2} {(5 \sqrt{3})}^{2} - \frac{1}{2} \cdot 0^{2}$

Step 6.8.3.6

Multiply $625$ by $9$ .

$- \frac{1}{75} (\frac{1}{4} \cdot 5625 - \frac{1}{4} \cdot 0^{4}) + \frac{1}{2} {(5 \sqrt{3})}^{2} - \frac{1}{2} \cdot 0^{2}$

Step 6.8.3.7

Combine $\frac{1}{4}$ and $5625$ .

$- \frac{1}{75} (\frac{5625}{4} - \frac{1}{4} \cdot 0^{4}) + \frac{1}{2} {(5 \sqrt{3})}^{2} - \frac{1}{2} \cdot 0^{2}$

Step 6.8.3.8

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

$- \frac{1}{75} (\frac{5625}{4} - \frac{1}{4} \cdot 0) + \frac{1}{2} {(5 \sqrt{3})}^{2} - \frac{1}{2} \cdot 0^{2}$

Step 6.8.3.9

Multiply $0$ by $- 1$ .

$- \frac{1}{75} (\frac{5625}{4} + 0 (\frac{1}{4})) + \frac{1}{2} {(5 \sqrt{3})}^{2} - \frac{1}{2} \cdot 0^{2}$

Step 6.8.3.10

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

$- \frac{1}{75} (\frac{5625}{4} + 0) + \frac{1}{2} {(5 \sqrt{3})}^{2} - \frac{1}{2} \cdot 0^{2}$

Step 6.8.3.11

Add $\frac{5625}{4}$ and $0$ .

$- \frac{1}{75} \cdot \frac{5625}{4} + \frac{1}{2} {(5 \sqrt{3})}^{2} - \frac{1}{2} \cdot 0^{2}$

Step 6.8.3.12

Multiply $\frac{5625}{4}$ by $\frac{1}{75}$ .

$- \frac{5625}{4 \cdot 75} + \frac{1}{2} {(5 \sqrt{3})}^{2} - \frac{1}{2} \cdot 0^{2}$

Step 6.8.3.13

Multiply $4$ by $75$ .

$- \frac{5625}{300} + \frac{1}{2} {(5 \sqrt{3})}^{2} - \frac{1}{2} \cdot 0^{2}$

Step 6.8.3.14

Cancel the common factor of $5625$ and $300$ .

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Step 6.8.3.14.1

Factor $75$ out of $5625$ .

$- \frac{75 (75)}{300} + \frac{1}{2} {(5 \sqrt{3})}^{2} - \frac{1}{2} \cdot 0^{2}$

Step 6.8.3.14.2

Cancel the common factors.

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Step 6.8.3.14.2.1

Factor $75$ out of $300$ .

$- \frac{75 \cdot 75}{75 \cdot 4} + \frac{1}{2} {(5 \sqrt{3})}^{2} - \frac{1}{2} \cdot 0^{2}$

Step 6.8.3.14.2.2

Cancel the common factor.

$- \frac{75 \cdot 75}{75 \cdot 4} + \frac{1}{2} {(5 \sqrt{3})}^{2} - \frac{1}{2} \cdot 0^{2}$

Step 6.8.3.14.2.3

Rewrite the expression.

$- \frac{75}{4} + \frac{1}{2} {(5 \sqrt{3})}^{2} - \frac{1}{2} \cdot 0^{2}$

Step 6.8.3.15

Factor $5$ out of $5 \sqrt{3}$ .

$- \frac{75}{4} + \frac{1}{2} {(5 (\sqrt{3}))}^{2} - \frac{1}{2} \cdot 0^{2}$

Step 6.8.3.16

Apply the product rule to $5 (\sqrt{3})$ .

$- \frac{75}{4} + \frac{1}{2} (5^{2} {\sqrt{3}}^{2}) - \frac{1}{2} \cdot 0^{2}$

Step 6.8.3.17

Raise $5$ to the power of $2$ .

$- \frac{75}{4} + \frac{1}{2} (25 {\sqrt{3}}^{2}) - \frac{1}{2} \cdot 0^{2}$

Step 6.8.3.18

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 6.8.3.18.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$- \frac{75}{4} + \frac{1}{2} (25 {(3^{\frac{1}{2}})}^{2}) - \frac{1}{2} \cdot 0^{2}$

Step 6.8.3.18.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- \frac{75}{4} + \frac{1}{2} (25 \cdot 3^{\frac{1}{2} \cdot 2}) - \frac{1}{2} \cdot 0^{2}$

Step 6.8.3.18.3

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

$- \frac{75}{4} + \frac{1}{2} (25 \cdot 3^{\frac{2}{2}}) - \frac{1}{2} \cdot 0^{2}$

Step 6.8.3.18.4

Cancel the common factor of $2$ .

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Step 6.8.3.18.4.1

Cancel the common factor.

$- \frac{75}{4} + \frac{1}{2} (25 \cdot 3^{\frac{2}{2}}) - \frac{1}{2} \cdot 0^{2}$

Step 6.8.3.18.4.2

Rewrite the expression.

$- \frac{75}{4} + \frac{1}{2} (25 \cdot 3^{1}) - \frac{1}{2} \cdot 0^{2}$

Step 6.8.3.18.5

Evaluate the exponent.

$- \frac{75}{4} + \frac{1}{2} (25 \cdot 3) - \frac{1}{2} \cdot 0^{2}$

Step 6.8.3.19

Multiply $25$ by $3$ .

$- \frac{75}{4} + \frac{1}{2} \cdot 75 - \frac{1}{2} \cdot 0^{2}$

Step 6.8.3.20

Combine $\frac{1}{2}$ and $75$ .

$- \frac{75}{4} + \frac{75}{2} - \frac{1}{2} \cdot 0^{2}$

Step 6.8.3.21

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

$- \frac{75}{4} + \frac{75}{2} - \frac{1}{2} \cdot 0$

Step 6.8.3.22

Multiply $0$ by $- 1$ .

$- \frac{75}{4} + \frac{75}{2} + 0 (\frac{1}{2})$

Step 6.8.3.23

Multiply $0$ by $\frac{1}{2}$ .

$- \frac{75}{4} + \frac{75}{2} + 0$

Step 6.8.3.24

Add $\frac{75}{2}$ and $0$ .

$- \frac{75}{4} + \frac{75}{2}$

Step 6.8.3.25

To write $\frac{75}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{75}{4} + \frac{75}{2} \cdot \frac{2}{2}$

Step 6.8.3.26

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.8.3.26.1

Multiply $\frac{75}{2}$ by $\frac{2}{2}$ .

$- \frac{75}{4} + \frac{75 \cdot 2}{2 \cdot 2}$

Step 6.8.3.26.2

Multiply $2$ by $2$ .

$- \frac{75}{4} + \frac{75 \cdot 2}{4}$

Step 6.8.3.27

Combine the numerators over the common denominator.

$\frac{- 75 + 75 \cdot 2}{4}$

Step 6.8.3.28

Simplify the numerator.

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Step 6.8.3.28.1

Multiply $75$ by $2$ .

$\frac{- 75 + 150}{4}$

Step 6.8.3.28.2

Add $- 75$ and $150$ .

$\frac{75}{4}$

Step 7

Add the areas $\frac{75}{4} + \frac{75}{4}$ .

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Step 7.1

Combine the numerators over the common denominator.

$\frac{75 + 75}{4}$

Step 7.2

Add $75$ and $75$ .

$\frac{150}{4}$

Step 7.3

Cancel the common factor of $150$ and $4$ .

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Step 7.3.1

Factor $2$ out of $150$ .

$\frac{2 (75)}{4}$

Step 7.3.2

Cancel the common factors.

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Step 7.3.2.1

Factor $2$ out of $4$ .

$\frac{2 \cdot 75}{2 \cdot 2}$

Step 7.3.2.2

Cancel the common factor.

$\frac{2 \cdot 75}{2 \cdot 2}$

Step 7.3.2.3

Rewrite the expression.

$\frac{75}{2}$

Step 8