Calculus Examples

$y = x^{2} - 6 x + 9$ ; $2 \leq x \leq 4$

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.

$x^{2} - 6 x + 9 = 0$

Step 1.2

Solve $x^{2} - 6 x + 9 = 0$ for $x$ .

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Step 1.2.1

Factor using the perfect square rule.

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Step 1.2.1.1

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

$x^{2} - 6 x + 3^{2} = 0$

Step 1.2.1.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$6 x = 2 \cdot x \cdot 3$

Step 1.2.1.3

Rewrite the polynomial.

$x^{2} - 2 \cdot x \cdot 3 + 3^{2} = 0$

Step 1.2.1.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 3$ .

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

Step 1.2.2

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

$x - 3 = 0$

Step 1.2.3

Add $3$ to both sides of the equation.

$x = 3$

Step 1.3

Substitute $3$ for $x$ .

$y = 0$

Step 1.4

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

$(3, 0)$

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.

$A r e a = \int_{2}^{3} x^{2} - 6 x + 9 d x - \int_{2}^{3} 0 d x$

Step 3

Integrate to find the area between $2$ and $3$ .

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Step 3.1

Combine the integrals into a single integral.

$\int_{2}^{3} x^{2} - 6 x + 9 - (0) d x$

Step 3.2

Subtract $0$ from $x^{2} - 6 x + 9$ .

$\int_{2}^{3} x^{2} - 6 x + 9 d x$

Step 3.3

Split the single integral into multiple integrals.

$\int_{2}^{3} x^{2} d x + \int_{2}^{3} - 6 x d x + \int_{2}^{3} 9 d x$

Step 3.4

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$\frac{1}{3} x^{3}]_{2}^{3} + \int_{2}^{3} - 6 x d x + \int_{2}^{3} 9 d x$

Step 3.5

Since $- 6$ is constant with respect to $x$ , move $- 6$ out of the integral.

$\frac{1}{3} x^{3}]_{2}^{3} - 6 \int_{2}^{3} x d x + \int_{2}^{3} 9 d x$

Step 3.6

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

$\frac{1}{3} x^{3}]_{2}^{3} - 6 (\frac{1}{2} x^{2}]_{2}^{3}) + \int_{2}^{3} 9 d x$

Step 3.7

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

$\frac{1}{3} x^{3}]_{2}^{3} - 6 (\frac{x^{2}}{2}]_{2}^{3}) + \int_{2}^{3} 9 d x$

Step 3.8

Apply the constant rule.

$\frac{1}{3} x^{3}]_{2}^{3} - 6 (\frac{x^{2}}{2}]_{2}^{3}) + 9 x]_{2}^{3}$

Step 3.9

Simplify the answer.

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Step 3.9.1

Combine $\frac{1}{3} x^{3}]_{2}^{3}$ and $9 x]_{2}^{3}$ .

$\frac{1}{3} x^{3} + 9 x]_{2}^{3} - 6 (\frac{x^{2}}{2}]_{2}^{3})$

Step 3.9.2

Substitute and simplify.

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Step 3.9.2.1

Evaluate $\frac{1}{3} x^{3} + 9 x$ at $3$ and at $2$ .

$(\frac{1}{3} \cdot 3^{3} + 9 \cdot 3) - (\frac{1}{3} \cdot 2^{3} + 9 \cdot 2) - 6 (\frac{x^{2}}{2}]_{2}^{3})$

Step 3.9.2.2

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

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

Step 3.9.2.3

Simplify.

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Step 3.9.2.3.1

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

$\frac{1}{3} \cdot 27 + 9 \cdot 3 - (\frac{1}{3} \cdot 2^{3} + 9 \cdot 2) - 6 ((\frac{3^{2}}{2}) - \frac{2^{2}}{2})$

Step 3.9.2.3.2

Combine $\frac{1}{3}$ and $27$ .

$\frac{27}{3} + 9 \cdot 3 - (\frac{1}{3} \cdot 2^{3} + 9 \cdot 2) - 6 ((\frac{3^{2}}{2}) - \frac{2^{2}}{2})$

Step 3.9.2.3.3

Cancel the common factor of $27$ and $3$ .

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Step 3.9.2.3.3.1

Factor $3$ out of $27$ .

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

Step 3.9.2.3.3.2

Cancel the common factors.

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Step 3.9.2.3.3.2.1

Factor $3$ out of $3$ .

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

Step 3.9.2.3.3.2.2

Cancel the common factor.

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

Step 3.9.2.3.3.2.3

Rewrite the expression.

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

Step 3.9.2.3.3.2.4

Divide $9$ by $1$ .

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

Step 3.9.2.3.4

Multiply $9$ by $3$ .

$9 + 27 - (\frac{1}{3} \cdot 2^{3} + 9 \cdot 2) - 6 ((\frac{3^{2}}{2}) - \frac{2^{2}}{2})$

Step 3.9.2.3.5

Add $9$ and $27$ .

$36 - (\frac{1}{3} \cdot 2^{3} + 9 \cdot 2) - 6 ((\frac{3^{2}}{2}) - \frac{2^{2}}{2})$

Step 3.9.2.3.6

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

$36 - (\frac{1}{3} \cdot 8 + 9 \cdot 2) - 6 ((\frac{3^{2}}{2}) - \frac{2^{2}}{2})$

Step 3.9.2.3.7

Combine $\frac{1}{3}$ and $8$ .

$36 - (\frac{8}{3} + 9 \cdot 2) - 6 ((\frac{3^{2}}{2}) - \frac{2^{2}}{2})$

Step 3.9.2.3.8

Multiply $9$ by $2$ .

$36 - (\frac{8}{3} + 18) - 6 ((\frac{3^{2}}{2}) - \frac{2^{2}}{2})$

Step 3.9.2.3.9

To write $18$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$36 - (\frac{8}{3} + 18 \cdot \frac{3}{3}) - 6 ((\frac{3^{2}}{2}) - \frac{2^{2}}{2})$

Step 3.9.2.3.10

Combine $18$ and $\frac{3}{3}$ .

$36 - (\frac{8}{3} + \frac{18 \cdot 3}{3}) - 6 ((\frac{3^{2}}{2}) - \frac{2^{2}}{2})$

Step 3.9.2.3.11

Combine the numerators over the common denominator.

$36 - \frac{8 + 18 \cdot 3}{3} - 6 ((\frac{3^{2}}{2}) - \frac{2^{2}}{2})$

Step 3.9.2.3.12

Simplify the numerator.

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Step 3.9.2.3.12.1

Multiply $18$ by $3$ .

$36 - \frac{8 + 54}{3} - 6 ((\frac{3^{2}}{2}) - \frac{2^{2}}{2})$

Step 3.9.2.3.12.2

Add $8$ and $54$ .

$36 - \frac{62}{3} - 6 ((\frac{3^{2}}{2}) - \frac{2^{2}}{2})$

Step 3.9.2.3.13

To write $36$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$36 \cdot \frac{3}{3} - \frac{62}{3} - 6 ((\frac{3^{2}}{2}) - \frac{2^{2}}{2})$

Step 3.9.2.3.14

Combine $36$ and $\frac{3}{3}$ .

$\frac{36 \cdot 3}{3} - \frac{62}{3} - 6 ((\frac{3^{2}}{2}) - \frac{2^{2}}{2})$

Step 3.9.2.3.15

Combine the numerators over the common denominator.

$\frac{36 \cdot 3 - 62}{3} - 6 ((\frac{3^{2}}{2}) - \frac{2^{2}}{2})$

Step 3.9.2.3.16

Simplify the numerator.

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Step 3.9.2.3.16.1

Multiply $36$ by $3$ .

$\frac{108 - 62}{3} - 6 ((\frac{3^{2}}{2}) - \frac{2^{2}}{2})$

Step 3.9.2.3.16.2

Subtract $62$ from $108$ .

$\frac{46}{3} - 6 ((\frac{3^{2}}{2}) - \frac{2^{2}}{2})$

Step 3.9.2.3.17

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

$\frac{46}{3} - 6 (\frac{9}{2} - \frac{2^{2}}{2})$

Step 3.9.2.3.18

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

$\frac{46}{3} - 6 (\frac{9}{2} - \frac{4}{2})$

Step 3.9.2.3.19

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

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Step 3.9.2.3.19.1

Factor $2$ out of $4$ .

$\frac{46}{3} - 6 (\frac{9}{2} - \frac{2 \cdot 2}{2})$

Step 3.9.2.3.19.2

Cancel the common factors.

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Step 3.9.2.3.19.2.1

Factor $2$ out of $2$ .

$\frac{46}{3} - 6 (\frac{9}{2} - \frac{2 \cdot 2}{2 (1)})$

Step 3.9.2.3.19.2.2

Cancel the common factor.

$\frac{46}{3} - 6 (\frac{9}{2} - \frac{2 \cdot 2}{2 \cdot 1})$

Step 3.9.2.3.19.2.3

Rewrite the expression.

$\frac{46}{3} - 6 (\frac{9}{2} - \frac{2}{1})$

Step 3.9.2.3.19.2.4

Divide $2$ by $1$ .

$\frac{46}{3} - 6 (\frac{9}{2} - 1 \cdot 2)$

Step 3.9.2.3.20

Multiply $- 1$ by $2$ .

$\frac{46}{3} - 6 (\frac{9}{2} - 2)$

Step 3.9.2.3.21

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

$\frac{46}{3} - 6 (\frac{9}{2} - 2 \cdot \frac{2}{2})$

Step 3.9.2.3.22

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

$\frac{46}{3} - 6 (\frac{9}{2} + \frac{- 2 \cdot 2}{2})$

Step 3.9.2.3.23

Combine the numerators over the common denominator.

$\frac{46}{3} - 6 \frac{9 - 2 \cdot 2}{2}$

Step 3.9.2.3.24

Simplify the numerator.

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Step 3.9.2.3.24.1

Multiply $- 2$ by $2$ .

$\frac{46}{3} - 6 \frac{9 - 4}{2}$

Step 3.9.2.3.24.2

Subtract $4$ from $9$ .

$\frac{46}{3} - 6 (\frac{5}{2})$

Step 3.9.2.3.25

Combine $- 6$ and $\frac{5}{2}$ .

$\frac{46}{3} + \frac{- 6 \cdot 5}{2}$

Step 3.9.2.3.26

Multiply $- 6$ by $5$ .

$\frac{46}{3} + \frac{- 30}{2}$

Step 3.9.2.3.27

Cancel the common factor of $- 30$ and $2$ .

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Step 3.9.2.3.27.1

Factor $2$ out of $- 30$ .

$\frac{46}{3} + \frac{2 \cdot - 15}{2}$

Step 3.9.2.3.27.2

Cancel the common factors.

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Step 3.9.2.3.27.2.1

Factor $2$ out of $2$ .

$\frac{46}{3} + \frac{2 \cdot - 15}{2 (1)}$

Step 3.9.2.3.27.2.2

Cancel the common factor.

$\frac{46}{3} + \frac{2 \cdot - 15}{2 \cdot 1}$

Step 3.9.2.3.27.2.3

Rewrite the expression.

$\frac{46}{3} + \frac{- 15}{1}$

Step 3.9.2.3.27.2.4

Divide $- 15$ by $1$ .

$\frac{46}{3} - 15$

Step 3.9.2.3.28

To write $- 15$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{46}{3} - 15 \cdot \frac{3}{3}$

Step 3.9.2.3.29

Combine $- 15$ and $\frac{3}{3}$ .

$\frac{46}{3} + \frac{- 15 \cdot 3}{3}$

Step 3.9.2.3.30

Combine the numerators over the common denominator.

$\frac{46 - 15 \cdot 3}{3}$

Step 3.9.2.3.31

Simplify the numerator.

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Step 3.9.2.3.31.1

Multiply $- 15$ by $3$ .

$\frac{46 - 45}{3}$

Step 3.9.2.3.31.2

Subtract $45$ from $46$ .

$\frac{1}{3}$

Step 4

$A r e a = \int_{3}^{4} x^{2} - 6 x + 9 d x - \int_{3}^{4} 0 d x$

Step 5

Integrate to find the area between $3$ and $4$ .

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Step 5.1

Combine the integrals into a single integral.

$\int_{3}^{4} x^{2} - 6 x + 9 - (0) d x$

Step 5.2

Subtract $0$ from $x^{2} - 6 x + 9$ .

$\int_{3}^{4} x^{2} - 6 x + 9 d x$

Step 5.3

Split the single integral into multiple integrals.

$\int_{3}^{4} x^{2} d x + \int_{3}^{4} - 6 x d x + \int_{3}^{4} 9 d x$

Step 5.4

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$\frac{1}{3} x^{3}]_{3}^{4} + \int_{3}^{4} - 6 x d x + \int_{3}^{4} 9 d x$

Step 5.5

Since $- 6$ is constant with respect to $x$ , move $- 6$ out of the integral.

$\frac{1}{3} x^{3}]_{3}^{4} - 6 \int_{3}^{4} x d x + \int_{3}^{4} 9 d x$

Step 5.6

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

$\frac{1}{3} x^{3}]_{3}^{4} - 6 (\frac{1}{2} x^{2}]_{3}^{4}) + \int_{3}^{4} 9 d x$

Step 5.7

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

$\frac{1}{3} x^{3}]_{3}^{4} - 6 (\frac{x^{2}}{2}]_{3}^{4}) + \int_{3}^{4} 9 d x$

Step 5.8

Apply the constant rule.

$\frac{1}{3} x^{3}]_{3}^{4} - 6 (\frac{x^{2}}{2}]_{3}^{4}) + 9 x]_{3}^{4}$

Step 5.9

Simplify the answer.

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Step 5.9.1

Combine $\frac{1}{3} x^{3}]_{3}^{4}$ and $9 x]_{3}^{4}$ .

$\frac{1}{3} x^{3} + 9 x]_{3}^{4} - 6 (\frac{x^{2}}{2}]_{3}^{4})$

Step 5.9.2

Substitute and simplify.

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Step 5.9.2.1

Evaluate $\frac{1}{3} x^{3} + 9 x$ at $4$ and at $3$ .

$(\frac{1}{3} \cdot 4^{3} + 9 \cdot 4) - (\frac{1}{3} \cdot 3^{3} + 9 \cdot 3) - 6 (\frac{x^{2}}{2}]_{3}^{4})$

Step 5.9.2.2

Evaluate $\frac{x^{2}}{2}$ at $4$ and at $3$ .

$(\frac{1}{3} \cdot 4^{3} + 9 \cdot 4) - (\frac{1}{3} \cdot 3^{3} + 9 \cdot 3) - 6 ((\frac{4^{2}}{2}) - \frac{3^{2}}{2})$

Step 5.9.2.3

Simplify.

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Step 5.9.2.3.1

Raise $4$ to the power of $3$ .

$\frac{1}{3} \cdot 64 + 9 \cdot 4 - (\frac{1}{3} \cdot 3^{3} + 9 \cdot 3) - 6 ((\frac{4^{2}}{2}) - \frac{3^{2}}{2})$

Step 5.9.2.3.2

Combine $\frac{1}{3}$ and $64$ .

$\frac{64}{3} + 9 \cdot 4 - (\frac{1}{3} \cdot 3^{3} + 9 \cdot 3) - 6 ((\frac{4^{2}}{2}) - \frac{3^{2}}{2})$

Step 5.9.2.3.3

Multiply $9$ by $4$ .

$\frac{64}{3} + 36 - (\frac{1}{3} \cdot 3^{3} + 9 \cdot 3) - 6 ((\frac{4^{2}}{2}) - \frac{3^{2}}{2})$

Step 5.9.2.3.4

To write $36$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{64}{3} + 36 \cdot \frac{3}{3} - (\frac{1}{3} \cdot 3^{3} + 9 \cdot 3) - 6 ((\frac{4^{2}}{2}) - \frac{3^{2}}{2})$

Step 5.9.2.3.5

Combine $36$ and $\frac{3}{3}$ .

$\frac{64}{3} + \frac{36 \cdot 3}{3} - (\frac{1}{3} \cdot 3^{3} + 9 \cdot 3) - 6 ((\frac{4^{2}}{2}) - \frac{3^{2}}{2})$

Step 5.9.2.3.6

Combine the numerators over the common denominator.

$\frac{64 + 36 \cdot 3}{3} - (\frac{1}{3} \cdot 3^{3} + 9 \cdot 3) - 6 ((\frac{4^{2}}{2}) - \frac{3^{2}}{2})$

Step 5.9.2.3.7

Simplify the numerator.

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Step 5.9.2.3.7.1

Multiply $36$ by $3$ .

$\frac{64 + 108}{3} - (\frac{1}{3} \cdot 3^{3} + 9 \cdot 3) - 6 ((\frac{4^{2}}{2}) - \frac{3^{2}}{2})$

Step 5.9.2.3.7.2

Add $64$ and $108$ .

$\frac{172}{3} - (\frac{1}{3} \cdot 3^{3} + 9 \cdot 3) - 6 ((\frac{4^{2}}{2}) - \frac{3^{2}}{2})$

Step 5.9.2.3.8

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

$\frac{172}{3} - (\frac{1}{3} \cdot 27 + 9 \cdot 3) - 6 ((\frac{4^{2}}{2}) - \frac{3^{2}}{2})$

Step 5.9.2.3.9

Combine $\frac{1}{3}$ and $27$ .

$\frac{172}{3} - (\frac{27}{3} + 9 \cdot 3) - 6 ((\frac{4^{2}}{2}) - \frac{3^{2}}{2})$

Step 5.9.2.3.10

Cancel the common factor of $27$ and $3$ .

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Step 5.9.2.3.10.1

Factor $3$ out of $27$ .

$\frac{172}{3} - (\frac{3 \cdot 9}{3} + 9 \cdot 3) - 6 ((\frac{4^{2}}{2}) - \frac{3^{2}}{2})$

Step 5.9.2.3.10.2

Cancel the common factors.

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Step 5.9.2.3.10.2.1

Factor $3$ out of $3$ .

$\frac{172}{3} - (\frac{3 \cdot 9}{3 (1)} + 9 \cdot 3) - 6 ((\frac{4^{2}}{2}) - \frac{3^{2}}{2})$

Step 5.9.2.3.10.2.2

Cancel the common factor.

$\frac{172}{3} - (\frac{3 \cdot 9}{3 \cdot 1} + 9 \cdot 3) - 6 ((\frac{4^{2}}{2}) - \frac{3^{2}}{2})$

Step 5.9.2.3.10.2.3

Rewrite the expression.

$\frac{172}{3} - (\frac{9}{1} + 9 \cdot 3) - 6 ((\frac{4^{2}}{2}) - \frac{3^{2}}{2})$

Step 5.9.2.3.10.2.4

Divide $9$ by $1$ .

$\frac{172}{3} - (9 + 9 \cdot 3) - 6 ((\frac{4^{2}}{2}) - \frac{3^{2}}{2})$

Step 5.9.2.3.11

Multiply $9$ by $3$ .

$\frac{172}{3} - (9 + 27) - 6 ((\frac{4^{2}}{2}) - \frac{3^{2}}{2})$

Step 5.9.2.3.12

Add $9$ and $27$ .

$\frac{172}{3} - 1 \cdot 36 - 6 ((\frac{4^{2}}{2}) - \frac{3^{2}}{2})$

Step 5.9.2.3.13

Multiply $- 1$ by $36$ .

$\frac{172}{3} - 36 - 6 ((\frac{4^{2}}{2}) - \frac{3^{2}}{2})$

Step 5.9.2.3.14

To write $- 36$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{172}{3} - 36 \cdot \frac{3}{3} - 6 ((\frac{4^{2}}{2}) - \frac{3^{2}}{2})$

Step 5.9.2.3.15

Combine $- 36$ and $\frac{3}{3}$ .

$\frac{172}{3} + \frac{- 36 \cdot 3}{3} - 6 ((\frac{4^{2}}{2}) - \frac{3^{2}}{2})$

Step 5.9.2.3.16

Combine the numerators over the common denominator.

$\frac{172 - 36 \cdot 3}{3} - 6 ((\frac{4^{2}}{2}) - \frac{3^{2}}{2})$

Step 5.9.2.3.17

Simplify the numerator.

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Step 5.9.2.3.17.1

Multiply $- 36$ by $3$ .

$\frac{172 - 108}{3} - 6 ((\frac{4^{2}}{2}) - \frac{3^{2}}{2})$

Step 5.9.2.3.17.2

Subtract $108$ from $172$ .

$\frac{64}{3} - 6 ((\frac{4^{2}}{2}) - \frac{3^{2}}{2})$

Step 5.9.2.3.18

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

$\frac{64}{3} - 6 (\frac{16}{2} - \frac{3^{2}}{2})$

Step 5.9.2.3.19

Cancel the common factor of $16$ and $2$ .

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Step 5.9.2.3.19.1

Factor $2$ out of $16$ .

$\frac{64}{3} - 6 (\frac{2 \cdot 8}{2} - \frac{3^{2}}{2})$

Step 5.9.2.3.19.2

Cancel the common factors.

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Step 5.9.2.3.19.2.1

Factor $2$ out of $2$ .

$\frac{64}{3} - 6 (\frac{2 \cdot 8}{2 (1)} - \frac{3^{2}}{2})$

Step 5.9.2.3.19.2.2

Cancel the common factor.

$\frac{64}{3} - 6 (\frac{2 \cdot 8}{2 \cdot 1} - \frac{3^{2}}{2})$

Step 5.9.2.3.19.2.3

Rewrite the expression.

$\frac{64}{3} - 6 (\frac{8}{1} - \frac{3^{2}}{2})$

Step 5.9.2.3.19.2.4

Divide $8$ by $1$ .

$\frac{64}{3} - 6 (8 - \frac{3^{2}}{2})$

Step 5.9.2.3.20

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

$\frac{64}{3} - 6 (8 - \frac{9}{2})$

Step 5.9.2.3.21

To write $8$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{64}{3} - 6 (8 \cdot \frac{2}{2} - \frac{9}{2})$

Step 5.9.2.3.22

Combine $8$ and $\frac{2}{2}$ .

$\frac{64}{3} - 6 (\frac{8 \cdot 2}{2} - \frac{9}{2})$

Step 5.9.2.3.23

Combine the numerators over the common denominator.

$\frac{64}{3} - 6 \frac{8 \cdot 2 - 9}{2}$

Step 5.9.2.3.24

Simplify the numerator.

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Step 5.9.2.3.24.1

Multiply $8$ by $2$ .

$\frac{64}{3} - 6 \frac{16 - 9}{2}$

Step 5.9.2.3.24.2

Subtract $9$ from $16$ .

$\frac{64}{3} - 6 (\frac{7}{2})$

Step 5.9.2.3.25

Combine $- 6$ and $\frac{7}{2}$ .

$\frac{64}{3} + \frac{- 6 \cdot 7}{2}$

Step 5.9.2.3.26

Multiply $- 6$ by $7$ .

$\frac{64}{3} + \frac{- 42}{2}$

Step 5.9.2.3.27

Cancel the common factor of $- 42$ and $2$ .

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Step 5.9.2.3.27.1

Factor $2$ out of $- 42$ .

$\frac{64}{3} + \frac{2 \cdot - 21}{2}$

Step 5.9.2.3.27.2

Cancel the common factors.

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Step 5.9.2.3.27.2.1

Factor $2$ out of $2$ .

$\frac{64}{3} + \frac{2 \cdot - 21}{2 (1)}$

Step 5.9.2.3.27.2.2

Cancel the common factor.

$\frac{64}{3} + \frac{2 \cdot - 21}{2 \cdot 1}$

Step 5.9.2.3.27.2.3

Rewrite the expression.

$\frac{64}{3} + \frac{- 21}{1}$

Step 5.9.2.3.27.2.4

Divide $- 21$ by $1$ .

$\frac{64}{3} - 21$

Step 5.9.2.3.28

To write $- 21$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{64}{3} - 21 \cdot \frac{3}{3}$

Step 5.9.2.3.29

Combine $- 21$ and $\frac{3}{3}$ .

$\frac{64}{3} + \frac{- 21 \cdot 3}{3}$

Step 5.9.2.3.30

Combine the numerators over the common denominator.

$\frac{64 - 21 \cdot 3}{3}$

Step 5.9.2.3.31

Simplify the numerator.

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Step 5.9.2.3.31.1

Multiply $- 21$ by $3$ .

$\frac{64 - 63}{3}$

Step 5.9.2.3.31.2

Subtract $63$ from $64$ .

$\frac{1}{3}$

Step 6

Add the areas $\frac{1}{3} + \frac{1}{3}$ .

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Step 6.1

Combine the numerators over the common denominator.

$\frac{1 + 1}{3}$

Step 6.2

Add $1$ and $1$ .

$\frac{2}{3}$

Step 7