Calculus Examples

$y = \sqrt{x}$ , $x = 16$ , $y = \frac{1}{3} x$

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.

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

Step 1.2

Solve $\sqrt{x} = \frac{1}{3} x$ for $x$ .

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Step 1.2.1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{x}}^{2} = {(\frac{1}{3} x)}^{2}$

Step 1.2.2

Simplify each side of the equation.

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Step 1.2.2.1

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

${(x^{\frac{1}{2}})}^{2} = {(\frac{1}{3} x)}^{2}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Simplify ${(x^{\frac{1}{2}})}^{2}$ .

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Step 1.2.2.2.1.1

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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Step 1.2.2.2.1.1.1

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

$x^{\frac{1}{2} \cdot 2} = {(\frac{1}{3} x)}^{2}$

Step 1.2.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 1.2.2.2.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = {(\frac{1}{3} x)}^{2}$

Step 1.2.2.2.1.1.2.2

Rewrite the expression.

$x^{1} = {(\frac{1}{3} x)}^{2}$

Step 1.2.2.2.1.2

Simplify.

$x = {(\frac{1}{3} x)}^{2}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Simplify ${(\frac{1}{3} x)}^{2}$ .

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Step 1.2.2.3.1.1

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

$x = {(\frac{x}{3})}^{2}$

Step 1.2.2.3.1.2

Simplify the expression.

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Step 1.2.2.3.1.2.1

Apply the product rule to $\frac{x}{3}$ .

$x = \frac{x^{2}}{3^{2}}$

Step 1.2.2.3.1.2.2

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

$x = \frac{x^{2}}{9}$

Step 1.2.3

Solve for $x$ .

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Step 1.2.3.1

Multiply both sides by $9$ .

$x \cdot 9 = \frac{x^{2}}{9} \cdot 9$

Step 1.2.3.2

Simplify.

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Step 1.2.3.2.1

Simplify the left side.

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Step 1.2.3.2.1.1

Move $9$ to the left of $x$ .

$9 x = \frac{x^{2}}{9} \cdot 9$

Step 1.2.3.2.2

Simplify the right side.

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Step 1.2.3.2.2.1

Cancel the common factor of $9$ .

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Step 1.2.3.2.2.1.1

Cancel the common factor.

$9 x = \frac{x^{2}}{9} \cdot 9$

Step 1.2.3.2.2.1.2

Rewrite the expression.

$9 x = x^{2}$

Step 1.2.3.3

Solve for $x$ .

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Step 1.2.3.3.1

Subtract $x^{2}$ from both sides of the equation.

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

Step 1.2.3.3.2

Factor the left side of the equation.

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Step 1.2.3.3.2.1

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$9 u - u^{2} = 0$

Step 1.2.3.3.2.2

Factor $u$ out of $9 u - u^{2}$ .

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Step 1.2.3.3.2.2.1

Factor $u$ out of $9 u$ .

$u \cdot 9 - u^{2} = 0$

Step 1.2.3.3.2.2.2

Factor $u$ out of $- u^{2}$ .

$u \cdot 9 + u (- u) = 0$

Step 1.2.3.3.2.2.3

Factor $u$ out of $u \cdot 9 + u (- u)$ .

$u (9 - u) = 0$

Step 1.2.3.3.2.3

Replace all occurrences of $u$ with $x$ .

$x (9 - x) = 0$

Step 1.2.3.3.3

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

$x = 0$

$9 - x = 0 + y = \frac{1}{3} x$

Step 1.2.3.3.4

Set $x$ equal to $0$ .

$x = 0$

Step 1.2.3.3.5

Set $9 - x$ equal to $0$ and solve for $x$ .

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Step 1.2.3.3.5.1

Set $9 - x$ equal to $0$ .

$9 - x = 0$

Step 1.2.3.3.5.2

Solve $9 - x = 0$ for $x$ .

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Step 1.2.3.3.5.2.1

Subtract $9$ from both sides of the equation.

$- x = - 9$

Step 1.2.3.3.5.2.2

Divide each term in $- x = - 9$ by $- 1$ and simplify.

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Step 1.2.3.3.5.2.2.1

Divide each term in $- x = - 9$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 9}{- 1}$

Step 1.2.3.3.5.2.2.2

Simplify the left side.

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Step 1.2.3.3.5.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 9}{- 1}$

Step 1.2.3.3.5.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 9}{- 1}$

Step 1.2.3.3.5.2.2.3

Simplify the right side.

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Step 1.2.3.3.5.2.2.3.1

Divide $- 9$ by $- 1$ .

$x = 9$

Step 1.2.3.3.6

The final solution is all the values that make $x (9 - x) = 0$ true.

$x = 0, 9$

Step 1.3

Evaluate $y$ when $x = 0$ .

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Step 1.3.1

Substitute $0$ for $x$ .

$y = \frac{1}{3} \cdot (0)$

Step 1.3.2

Substitute $0$ for $x$ in $y = \frac{1}{3} \cdot (0)$ and solve for $y$ .

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Step 1.3.2.1

Multiply $\frac{1}{3}$ by $0$ .

$y = \frac{1}{3} \cdot 0$

Step 1.3.2.2

Multiply $\frac{1}{3}$ by $0$ .

$y = 0$

Step 1.4

Evaluate $y$ when $x = 9$ .

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Step 1.4.1

Substitute $9$ for $x$ .

$y = \frac{1}{3} \cdot (9)$

Step 1.4.2

Substitute $9$ for $x$ in $y = \frac{1}{3} \cdot (9)$ and solve for $y$ .

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Step 1.4.2.1

Multiply $\frac{1}{3}$ by $9$ .

$y = \frac{1}{3} \cdot 9$

Step 1.4.2.2

Cancel the common factor of $3$ .

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Step 1.4.2.2.1

Factor $3$ out of $9$ .

$y = \frac{1}{3} \cdot (3 (3))$

Step 1.4.2.2.2

Cancel the common factor.

$y = \frac{1}{3} \cdot (3 \cdot 3)$

Step 1.4.2.2.3

Rewrite the expression.

$y = 3$

Step 1.5

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

$(0, 0)$

$(9, 3)$

$(0, 0)$

$(9, 3)$

Step 2

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

$y = \frac{x}{3}$

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.

$A r e a = \int_{0}^{9} \sqrt{x} d x - \int_{0}^{9} \frac{x}{3} d x$

Step 4

Integrate to find the area between $0$ and $9$ .

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Step 4.1

Combine the integrals into a single integral.

$\int_{0}^{9} \sqrt{x} - (\frac{x}{3}) d x$

Step 4.2

Multiply $- 1$ by $\frac{x}{3}$ .

$\int_{0}^{9} \sqrt{x} - \frac{x}{3} d x$

Step 4.3

Split the single integral into multiple integrals.

$\int_{0}^{9} \sqrt{x} d x + \int_{0}^{9} - \frac{x}{3} d x$

Step 4.4

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

$\int_{0}^{9} x^{\frac{1}{2}} d x + \int_{0}^{9} - \frac{x}{3} d x$

Step 4.5

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

$\frac{2}{3} x^{\frac{3}{2}}]_{0}^{9} + \int_{0}^{9} - \frac{x}{3} d x$

Step 4.6

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

$\frac{2}{3} x^{\frac{3}{2}}]_{0}^{9} - \int_{0}^{9} \frac{x}{3} d x$

Step 4.7

Since $\frac{1}{3}$ is constant with respect to $x$ , move $\frac{1}{3}$ out of the integral.

$\frac{2}{3} x^{\frac{3}{2}}]_{0}^{9} - (\frac{1}{3} \int_{0}^{9} x d x)$

Step 4.8

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

$\frac{2}{3} x^{\frac{3}{2}}]_{0}^{9} - \frac{1}{3} \frac{1}{2} x^{2}]_{0}^{9}$

Step 4.9

Substitute and simplify.

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Step 4.9.1

Evaluate $\frac{2}{3} x^{\frac{3}{2}}$ at $9$ and at $0$ .

$(\frac{2}{3} \cdot 9^{\frac{3}{2}}) - \frac{2}{3} \cdot 0^{\frac{3}{2}} - \frac{1}{3} \frac{1}{2} x^{2}]_{0}^{9}$

Step 4.9.2

Evaluate $\frac{1}{2} x^{2}$ at $9$ and at $0$ .

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

Step 4.9.3

Simplify.

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Step 4.9.3.1

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

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

Step 4.9.3.2

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

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

Step 4.9.3.3

Cancel the common factor of $2$ .

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Step 4.9.3.3.1

Cancel the common factor.

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

Step 4.9.3.3.2

Rewrite the expression.

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

Step 4.9.3.4

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

$\frac{2}{3} \cdot 27 - \frac{2}{3} \cdot 0^{\frac{3}{2}} - \frac{1}{3} (\frac{1}{2} \cdot 9^{2} - \frac{1}{2} \cdot 0^{2})$

Step 4.9.3.5

Combine $\frac{2}{3}$ and $27$ .

$\frac{2 \cdot 27}{3} - \frac{2}{3} \cdot 0^{\frac{3}{2}} - \frac{1}{3} (\frac{1}{2} \cdot 9^{2} - \frac{1}{2} \cdot 0^{2})$

Step 4.9.3.6

Multiply $2$ by $27$ .

$\frac{54}{3} - \frac{2}{3} \cdot 0^{\frac{3}{2}} - \frac{1}{3} (\frac{1}{2} \cdot 9^{2} - \frac{1}{2} \cdot 0^{2})$

Step 4.9.3.7

Cancel the common factor of $54$ and $3$ .

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Step 4.9.3.7.1

Factor $3$ out of $54$ .

$\frac{3 \cdot 18}{3} - \frac{2}{3} \cdot 0^{\frac{3}{2}} - \frac{1}{3} (\frac{1}{2} \cdot 9^{2} - \frac{1}{2} \cdot 0^{2})$

Step 4.9.3.7.2

Cancel the common factors.

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Step 4.9.3.7.2.1

Factor $3$ out of $3$ .

$\frac{3 \cdot 18}{3 (1)} - \frac{2}{3} \cdot 0^{\frac{3}{2}} - \frac{1}{3} (\frac{1}{2} \cdot 9^{2} - \frac{1}{2} \cdot 0^{2})$

Step 4.9.3.7.2.2

Cancel the common factor.

$\frac{3 \cdot 18}{3 \cdot 1} - \frac{2}{3} \cdot 0^{\frac{3}{2}} - \frac{1}{3} (\frac{1}{2} \cdot 9^{2} - \frac{1}{2} \cdot 0^{2})$

Step 4.9.3.7.2.3

Rewrite the expression.

$\frac{18}{1} - \frac{2}{3} \cdot 0^{\frac{3}{2}} - \frac{1}{3} (\frac{1}{2} \cdot 9^{2} - \frac{1}{2} \cdot 0^{2})$

Step 4.9.3.7.2.4

Divide $18$ by $1$ .

$18 - \frac{2}{3} \cdot 0^{\frac{3}{2}} - \frac{1}{3} (\frac{1}{2} \cdot 9^{2} - \frac{1}{2} \cdot 0^{2})$

Step 4.9.3.8

Rewrite $0$ as $0^{2}$ .

$18 - \frac{2}{3} \cdot {(0^{2})}^{\frac{3}{2}} - \frac{1}{3} (\frac{1}{2} \cdot 9^{2} - \frac{1}{2} \cdot 0^{2})$

Step 4.9.3.9

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

$18 - \frac{2}{3} \cdot 0^{2 (\frac{3}{2})} - \frac{1}{3} (\frac{1}{2} \cdot 9^{2} - \frac{1}{2} \cdot 0^{2})$

Step 4.9.3.10

Cancel the common factor of $2$ .

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Step 4.9.3.10.1

Cancel the common factor.

$18 - \frac{2}{3} \cdot 0^{2 (\frac{3}{2})} - \frac{1}{3} (\frac{1}{2} \cdot 9^{2} - \frac{1}{2} \cdot 0^{2})$

Step 4.9.3.10.2

Rewrite the expression.

$18 - \frac{2}{3} \cdot 0^{3} - \frac{1}{3} (\frac{1}{2} \cdot 9^{2} - \frac{1}{2} \cdot 0^{2})$

Step 4.9.3.11

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

$18 - \frac{2}{3} \cdot 0 - \frac{1}{3} (\frac{1}{2} \cdot 9^{2} - \frac{1}{2} \cdot 0^{2})$

Step 4.9.3.12

Multiply $0$ by $- 1$ .

$18 + 0 (\frac{2}{3}) - \frac{1}{3} (\frac{1}{2} \cdot 9^{2} - \frac{1}{2} \cdot 0^{2})$

Step 4.9.3.13

Multiply $0$ by $\frac{2}{3}$ .

$18 + 0 - \frac{1}{3} (\frac{1}{2} \cdot 9^{2} - \frac{1}{2} \cdot 0^{2})$

Step 4.9.3.14

Add $18$ and $0$ .

$18 - \frac{1}{3} (\frac{1}{2} \cdot 9^{2} - \frac{1}{2} \cdot 0^{2})$

Step 4.9.3.15

Raise $9$ to the power of $2$ .

$18 - \frac{1}{3} (\frac{1}{2} \cdot 81 - \frac{1}{2} \cdot 0^{2})$

Step 4.9.3.16

Combine $\frac{1}{2}$ and $81$ .

$18 - \frac{1}{3} (\frac{81}{2} - \frac{1}{2} \cdot 0^{2})$

Step 4.9.3.17

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

$18 - \frac{1}{3} (\frac{81}{2} - \frac{1}{2} \cdot 0)$

Step 4.9.3.18

Multiply $0$ by $- 1$ .

$18 - \frac{1}{3} (\frac{81}{2} + 0 (\frac{1}{2}))$

Step 4.9.3.19

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

$18 - \frac{1}{3} (\frac{81}{2} + 0)$

Step 4.9.3.20

Add $\frac{81}{2}$ and $0$ .

$18 - \frac{1}{3} \cdot \frac{81}{2}$

Step 4.9.3.21

Multiply $\frac{81}{2}$ by $\frac{1}{3}$ .

$18 - \frac{81}{2 \cdot 3}$

Step 4.9.3.22

Multiply $2$ by $3$ .

$18 - \frac{81}{6}$

Step 4.9.3.23

Cancel the common factor of $81$ and $6$ .

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Step 4.9.3.23.1

Factor $3$ out of $81$ .

$18 - \frac{3 (27)}{6}$

Step 4.9.3.23.2

Cancel the common factors.

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Step 4.9.3.23.2.1

Factor $3$ out of $6$ .

$18 - \frac{3 \cdot 27}{3 \cdot 2}$

Step 4.9.3.23.2.2

Cancel the common factor.

$18 - \frac{3 \cdot 27}{3 \cdot 2}$

Step 4.9.3.23.2.3

Rewrite the expression.

$18 - \frac{27}{2}$

Step 4.9.3.24

To write $18$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$18 \cdot \frac{2}{2} - \frac{27}{2}$

Step 4.9.3.25

Combine $18$ and $\frac{2}{2}$ .

$\frac{18 \cdot 2}{2} - \frac{27}{2}$

Step 4.9.3.26

Combine the numerators over the common denominator.

$\frac{18 \cdot 2 - 27}{2}$

Step 4.9.3.27

Simplify the numerator.

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Step 4.9.3.27.1

Multiply $18$ by $2$ .

$\frac{36 - 27}{2}$

Step 4.9.3.27.2

Subtract $27$ from $36$ .

$\frac{9}{2}$

Step 5

$A r e a = \int_{9}^{16} \frac{x}{3} d x - \int_{9}^{16} \sqrt{x} d x$

Step 6

Integrate to find the area between $9$ and $16$ .

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Step 6.1

Combine the integrals into a single integral.

$\int_{9}^{16} \frac{x}{3} - (\sqrt{x}) d x$

Step 6.2

Multiply $- 1$ by $\sqrt{x}$ .

$\int_{9}^{16} \frac{x}{3} - \sqrt{x} d x$

Step 6.3

Split the single integral into multiple integrals.

$\int_{9}^{16} \frac{x}{3} d x + \int_{9}^{16} - \sqrt{x} d x$

Step 6.4

Since $\frac{1}{3}$ is constant with respect to $x$ , move $\frac{1}{3}$ out of the integral.

$\frac{1}{3} \int_{9}^{16} x d x + \int_{9}^{16} - \sqrt{x} d x$

Step 6.5

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

$\frac{1}{3} \frac{1}{2} x^{2}]_{9}^{16} + \int_{9}^{16} - \sqrt{x} d x$

Step 6.6

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

$\frac{1}{3} \frac{1}{2} x^{2}]_{9}^{16} - \int_{9}^{16} \sqrt{x} d x$

Step 6.7

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

$\frac{1}{3} \frac{1}{2} x^{2}]_{9}^{16} - \int_{9}^{16} x^{\frac{1}{2}} d x$

Step 6.8

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

$\frac{1}{3} \frac{1}{2} x^{2}]_{9}^{16} - (\frac{2}{3} x^{\frac{3}{2}}]_{9}^{16})$

Step 6.9

Simplify the answer.

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Step 6.9.1

Combine $\frac{2}{3}$ and $x^{\frac{3}{2}}$ .

$\frac{1}{3} \frac{1}{2} x^{2}]_{9}^{16} - (\frac{2 x^{\frac{3}{2}}}{3}]_{9}^{16})$

Step 6.9.2

Substitute and simplify.

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Step 6.9.2.1

Evaluate $\frac{1}{2} x^{2}$ at $16$ and at $9$ .

$\frac{1}{3} ((\frac{1}{2} \cdot 16^{2}) - \frac{1}{2} \cdot 9^{2}) - (\frac{2 x^{\frac{3}{2}}}{3}]_{9}^{16})$

Step 6.9.2.2

Evaluate $\frac{2 x^{\frac{3}{2}}}{3}$ at $16$ and at $9$ .

$\frac{1}{3} (\frac{1}{2} \cdot 16^{2} - \frac{1}{2} \cdot 9^{2}) - (\frac{2 \cdot 16^{\frac{3}{2}}}{3} - \frac{2 \cdot 9^{\frac{3}{2}}}{3})$

Step 6.9.2.3

Simplify.

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Step 6.9.2.3.1

Raise $16$ to the power of $2$ .

$\frac{1}{3} (\frac{1}{2} \cdot 256 - \frac{1}{2} \cdot 9^{2}) - (\frac{2 \cdot 16^{\frac{3}{2}}}{3} - \frac{2 \cdot 9^{\frac{3}{2}}}{3})$

Step 6.9.2.3.2

Combine $\frac{1}{2}$ and $256$ .

$\frac{1}{3} (\frac{256}{2} - \frac{1}{2} \cdot 9^{2}) - (\frac{2 \cdot 16^{\frac{3}{2}}}{3} - \frac{2 \cdot 9^{\frac{3}{2}}}{3})$

Step 6.9.2.3.3

Cancel the common factor of $256$ and $2$ .

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Step 6.9.2.3.3.1

Factor $2$ out of $256$ .

$\frac{1}{3} (\frac{2 \cdot 128}{2} - \frac{1}{2} \cdot 9^{2}) - (\frac{2 \cdot 16^{\frac{3}{2}}}{3} - \frac{2 \cdot 9^{\frac{3}{2}}}{3})$

Step 6.9.2.3.3.2

Cancel the common factors.

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Step 6.9.2.3.3.2.1

Factor $2$ out of $2$ .

$\frac{1}{3} (\frac{2 \cdot 128}{2 (1)} - \frac{1}{2} \cdot 9^{2}) - (\frac{2 \cdot 16^{\frac{3}{2}}}{3} - \frac{2 \cdot 9^{\frac{3}{2}}}{3})$

Step 6.9.2.3.3.2.2

Cancel the common factor.

$\frac{1}{3} (\frac{2 \cdot 128}{2 \cdot 1} - \frac{1}{2} \cdot 9^{2}) - (\frac{2 \cdot 16^{\frac{3}{2}}}{3} - \frac{2 \cdot 9^{\frac{3}{2}}}{3})$

Step 6.9.2.3.3.2.3

Rewrite the expression.

$\frac{1}{3} (\frac{128}{1} - \frac{1}{2} \cdot 9^{2}) - (\frac{2 \cdot 16^{\frac{3}{2}}}{3} - \frac{2 \cdot 9^{\frac{3}{2}}}{3})$

Step 6.9.2.3.3.2.4

Divide $128$ by $1$ .

$\frac{1}{3} (128 - \frac{1}{2} \cdot 9^{2}) - (\frac{2 \cdot 16^{\frac{3}{2}}}{3} - \frac{2 \cdot 9^{\frac{3}{2}}}{3})$

Step 6.9.2.3.4

Raise $9$ to the power of $2$ .

$\frac{1}{3} (128 - \frac{1}{2} \cdot 81) - (\frac{2 \cdot 16^{\frac{3}{2}}}{3} - \frac{2 \cdot 9^{\frac{3}{2}}}{3})$

Step 6.9.2.3.5

Multiply $81$ by $- 1$ .

$\frac{1}{3} (128 - 81 (\frac{1}{2})) - (\frac{2 \cdot 16^{\frac{3}{2}}}{3} - \frac{2 \cdot 9^{\frac{3}{2}}}{3})$

Step 6.9.2.3.6

Combine $- 81$ and $\frac{1}{2}$ .

$\frac{1}{3} (128 + \frac{- 81}{2}) - (\frac{2 \cdot 16^{\frac{3}{2}}}{3} - \frac{2 \cdot 9^{\frac{3}{2}}}{3})$

Step 6.9.2.3.7

Move the negative in front of the fraction.

$\frac{1}{3} (128 - \frac{81}{2}) - (\frac{2 \cdot 16^{\frac{3}{2}}}{3} - \frac{2 \cdot 9^{\frac{3}{2}}}{3})$

Step 6.9.2.3.8

To write $128$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{3} (128 \cdot \frac{2}{2} - \frac{81}{2}) - (\frac{2 \cdot 16^{\frac{3}{2}}}{3} - \frac{2 \cdot 9^{\frac{3}{2}}}{3})$

Step 6.9.2.3.9

Combine $128$ and $\frac{2}{2}$ .

$\frac{1}{3} (\frac{128 \cdot 2}{2} - \frac{81}{2}) - (\frac{2 \cdot 16^{\frac{3}{2}}}{3} - \frac{2 \cdot 9^{\frac{3}{2}}}{3})$

Step 6.9.2.3.10

Combine the numerators over the common denominator.

$\frac{1}{3} \cdot \frac{128 \cdot 2 - 81}{2} - (\frac{2 \cdot 16^{\frac{3}{2}}}{3} - \frac{2 \cdot 9^{\frac{3}{2}}}{3})$

Step 6.9.2.3.11

Simplify the numerator.

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Step 6.9.2.3.11.1

Multiply $128$ by $2$ .

$\frac{1}{3} \cdot \frac{256 - 81}{2} - (\frac{2 \cdot 16^{\frac{3}{2}}}{3} - \frac{2 \cdot 9^{\frac{3}{2}}}{3})$

Step 6.9.2.3.11.2

Subtract $81$ from $256$ .

$\frac{1}{3} \cdot \frac{175}{2} - (\frac{2 \cdot 16^{\frac{3}{2}}}{3} - \frac{2 \cdot 9^{\frac{3}{2}}}{3})$

Step 6.9.2.3.12

Multiply $\frac{1}{3}$ by $\frac{175}{2}$ .

$\frac{175}{3 \cdot 2} - (\frac{2 \cdot 16^{\frac{3}{2}}}{3} - \frac{2 \cdot 9^{\frac{3}{2}}}{3})$

Step 6.9.2.3.13

Multiply $3$ by $2$ .

$\frac{175}{6} - (\frac{2 \cdot 16^{\frac{3}{2}}}{3} - \frac{2 \cdot 9^{\frac{3}{2}}}{3})$

Step 6.9.2.3.14

Rewrite $16$ as $4^{2}$ .

$\frac{175}{6} - (\frac{2 \cdot {(4^{2})}^{\frac{3}{2}}}{3} - \frac{2 \cdot 9^{\frac{3}{2}}}{3})$

Step 6.9.2.3.15

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

$\frac{175}{6} - (\frac{2 \cdot 4^{2 (\frac{3}{2})}}{3} - \frac{2 \cdot 9^{\frac{3}{2}}}{3})$

Step 6.9.2.3.16

Cancel the common factor of $2$ .

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Step 6.9.2.3.16.1

Cancel the common factor.

$\frac{175}{6} - (\frac{2 \cdot 4^{2 (\frac{3}{2})}}{3} - \frac{2 \cdot 9^{\frac{3}{2}}}{3})$

Step 6.9.2.3.16.2

Rewrite the expression.

$\frac{175}{6} - (\frac{2 \cdot 4^{3}}{3} - \frac{2 \cdot 9^{\frac{3}{2}}}{3})$

Step 6.9.2.3.17

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

$\frac{175}{6} - (\frac{2 \cdot 64}{3} - \frac{2 \cdot 9^{\frac{3}{2}}}{3})$

Step 6.9.2.3.18

Multiply $2$ by $64$ .

$\frac{175}{6} - (\frac{128}{3} - \frac{2 \cdot 9^{\frac{3}{2}}}{3})$

Step 6.9.2.3.19

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

$\frac{175}{6} - (\frac{128}{3} - \frac{2 \cdot {(3^{2})}^{\frac{3}{2}}}{3})$

Step 6.9.2.3.20

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

$\frac{175}{6} - (\frac{128}{3} - \frac{2 \cdot 3^{2 (\frac{3}{2})}}{3})$

Step 6.9.2.3.21

Cancel the common factor of $2$ .

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Step 6.9.2.3.21.1

Cancel the common factor.

$\frac{175}{6} - (\frac{128}{3} - \frac{2 \cdot 3^{2 (\frac{3}{2})}}{3})$

Step 6.9.2.3.21.2

Rewrite the expression.

$\frac{175}{6} - (\frac{128}{3} - \frac{2 \cdot 3^{3}}{3})$

Step 6.9.2.3.22

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

$\frac{175}{6} - (\frac{128}{3} - \frac{2 \cdot 27}{3})$

Step 6.9.2.3.23

Multiply $2$ by $27$ .

$\frac{175}{6} - (\frac{128}{3} - \frac{54}{3})$

Step 6.9.2.3.24

Cancel the common factor of $54$ and $3$ .

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Step 6.9.2.3.24.1

Factor $3$ out of $54$ .

$\frac{175}{6} - (\frac{128}{3} - \frac{3 \cdot 18}{3})$

Step 6.9.2.3.24.2

Cancel the common factors.

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Step 6.9.2.3.24.2.1

Factor $3$ out of $3$ .

$\frac{175}{6} - (\frac{128}{3} - \frac{3 \cdot 18}{3 (1)})$

Step 6.9.2.3.24.2.2

Cancel the common factor.

$\frac{175}{6} - (\frac{128}{3} - \frac{3 \cdot 18}{3 \cdot 1})$

Step 6.9.2.3.24.2.3

Rewrite the expression.

$\frac{175}{6} - (\frac{128}{3} - \frac{18}{1})$

Step 6.9.2.3.24.2.4

Divide $18$ by $1$ .

$\frac{175}{6} - (\frac{128}{3} - 1 \cdot 18)$

Step 6.9.2.3.25

Multiply $- 1$ by $18$ .

$\frac{175}{6} - (\frac{128}{3} - 18)$

Step 6.9.2.3.26

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

$\frac{175}{6} - (\frac{128}{3} - 18 \cdot \frac{3}{3})$

Step 6.9.2.3.27

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

$\frac{175}{6} - (\frac{128}{3} + \frac{- 18 \cdot 3}{3})$

Step 6.9.2.3.28

Combine the numerators over the common denominator.

$\frac{175}{6} - \frac{128 - 18 \cdot 3}{3}$

Step 6.9.2.3.29

Simplify the numerator.

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Step 6.9.2.3.29.1

Multiply $- 18$ by $3$ .

$\frac{175}{6} - \frac{128 - 54}{3}$

Step 6.9.2.3.29.2

Subtract $54$ from $128$ .

$\frac{175}{6} - \frac{74}{3}$

Step 6.9.2.3.30

To write $- \frac{74}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{175}{6} - \frac{74}{3} \cdot \frac{2}{2}$

Step 6.9.2.3.31

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.9.2.3.31.1

Multiply $\frac{74}{3}$ by $\frac{2}{2}$ .

$\frac{175}{6} - \frac{74 \cdot 2}{3 \cdot 2}$

Step 6.9.2.3.31.2

Multiply $3$ by $2$ .

$\frac{175}{6} - \frac{74 \cdot 2}{6}$

Step 6.9.2.3.32

Combine the numerators over the common denominator.

$\frac{175 - 74 \cdot 2}{6}$

Step 6.9.2.3.33

Simplify the numerator.

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Step 6.9.2.3.33.1

Multiply $- 74$ by $2$ .

$\frac{175 - 148}{6}$

Step 6.9.2.3.33.2

Subtract $148$ from $175$ .

$\frac{27}{6}$

Step 6.9.2.3.34

Cancel the common factor of $27$ and $6$ .

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Step 6.9.2.3.34.1

Factor $3$ out of $27$ .

$\frac{3 (9)}{6}$

Step 6.9.2.3.34.2

Cancel the common factors.

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Step 6.9.2.3.34.2.1

Factor $3$ out of $6$ .

$\frac{3 \cdot 9}{3 \cdot 2}$

Step 6.9.2.3.34.2.2

Cancel the common factor.

$\frac{3 \cdot 9}{3 \cdot 2}$

Step 6.9.2.3.34.2.3

Rewrite the expression.

$\frac{9}{2}$

Step 7

Add the areas $\frac{9}{2} + \frac{9}{2}$ .

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Step 7.1

Combine the numerators over the common denominator.

$\frac{9 + 9}{2}$

Step 7.2

Simplify the expression.

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Step 7.2.1

Add $9$ and $9$ .

$\frac{18}{2}$

Step 7.2.2

Divide $18$ by $2$ .

$9$

Step 8