Calculus Examples

$x = 24 y^{2} - 8 y^{3}$ , $x = 6 y^{2} - 18 y$

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.

$24 y^{2} - 8 y^{3} = 6 y^{2} - 18 y$

Step 1.2

Solve $24 y^{2} - 8 y^{3} = 6 y^{2} - 18 y$ for $y$ .

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Step 1.2.1

Move all terms containing $y$ to the left side of the equation.

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Step 1.2.1.1

Subtract $6 y^{2}$ from both sides of the equation.

$24 y^{2} - 8 y^{3} - 6 y^{2} = - 18 y$

Step 1.2.1.2

Add $18 y$ to both sides of the equation.

$24 y^{2} - 8 y^{3} - 6 y^{2} + 18 y = 0$

Step 1.2.1.3

Subtract $6 y^{2}$ from $24 y^{2}$ .

$- 8 y^{3} + 18 y^{2} + 18 y = 0$

Step 1.2.2

Factor the left side of the equation.

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Step 1.2.2.1

Factor $- 2 y$ out of $- 8 y^{3} + 18 y^{2} + 18 y$ .

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Step 1.2.2.1.1

Factor $- 2 y$ out of $- 8 y^{3}$ .

$- 2 y (4 y^{2}) + 18 y^{2} + 18 y = 0$

Step 1.2.2.1.2

Factor $- 2 y$ out of $18 y^{2}$ .

$- 2 y (4 y^{2}) - 2 y (- 9 y) + 18 y = 0$

Step 1.2.2.1.3

Factor $- 2 y$ out of $18 y$ .

$- 2 y (4 y^{2}) - 2 y (- 9 y) - 2 y \cdot - 9 = 0$

Step 1.2.2.1.4

Factor $- 2 y$ out of $- 2 y (4 y^{2}) - 2 y (- 9 y)$ .

$- 2 y (4 y^{2} - 9 y) - 2 y \cdot - 9 = 0$

Step 1.2.2.1.5

Factor $- 2 y$ out of $- 2 y (4 y^{2} - 9 y) - 2 y (- 9)$ .

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

Step 1.2.2.2

Factor.

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Step 1.2.2.2.1

Factor by grouping.

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Step 1.2.2.2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 4 \cdot - 9 = - 36$ and whose sum is $b = - 9$ .

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Step 1.2.2.2.1.1.1

Factor $- 9$ out of $- 9 y$ .

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

Step 1.2.2.2.1.1.2

Rewrite $- 9$ as $3$ plus $- 12$

$- 2 y (4 y^{2} + (3 - 12) y - 9) = 0$

Step 1.2.2.2.1.1.3

Apply the distributive property.

$- 2 y (4 y^{2} + 3 y - 12 y - 9) = 0$

Step 1.2.2.2.1.2

Factor out the greatest common factor from each group.

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Step 1.2.2.2.1.2.1

Group the first two terms and the last two terms.

$- 2 y ((4 y^{2} + 3 y) - 12 y - 9) = 0$

Step 1.2.2.2.1.2.2

Factor out the greatest common factor (GCF) from each group.

$- 2 y (y (4 y + 3) - 3 (4 y + 3)) = 0$

Step 1.2.2.2.1.3

Factor the polynomial by factoring out the greatest common factor, $4 y + 3$ .

$- 2 y ((4 y + 3) (y - 3)) = 0$

Step 1.2.2.2.2

Remove unnecessary parentheses.

$- 2 y (4 y + 3) (y - 3) = 0$

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 $0$ .

$y = 0$

$4 y + 3 = 0$

$y - 3 = 0 + x = 6 y^{2} - 18 y$

Step 1.2.4

Set $y$ equal to $0$ .

$y = 0$

Step 1.2.5

Set $4 y + 3$ equal to $0$ and solve for $y$ .

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Step 1.2.5.1

Set $4 y + 3$ equal to $0$ .

$4 y + 3 = 0$

Step 1.2.5.2

Solve $4 y + 3 = 0$ for $y$ .

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Step 1.2.5.2.1

Subtract $3$ from both sides of the equation.

$4 y = - 3$

Step 1.2.5.2.2

Divide each term in $4 y = - 3$ by $4$ and simplify.

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Step 1.2.5.2.2.1

Divide each term in $4 y = - 3$ by $4$ .

$\frac{4 y}{4} = \frac{- 3}{4}$

Step 1.2.5.2.2.2

Simplify the left side.

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Step 1.2.5.2.2.2.1

Cancel the common factor of $4$ .

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Step 1.2.5.2.2.2.1.1

Cancel the common factor.

$\frac{4 y}{4} = \frac{- 3}{4}$

Step 1.2.5.2.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 3}{4}$

Step 1.2.5.2.2.3

Simplify the right side.

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Step 1.2.5.2.2.3.1

Move the negative in front of the fraction.

$y = - \frac{3}{4}$

Step 1.2.6

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

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Step 1.2.6.1

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

$y - 3 = 0$

Step 1.2.6.2

Add $3$ to both sides of the equation.

$y = 3$

Step 1.2.7

The final solution is all the values that make $- 2 y (4 y + 3) (y - 3) = 0$ true.

$y = 0, - \frac{3}{4}, 3$

Step 1.3

Evaluate $x$ when $y = 0$ .

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Step 1.3.1

Substitute $0$ for $y$ .

$x = 6 {(0)}^{2} - 18 \cdot 0$

Step 1.3.2

Substitute $0$ for $y$ in $x = 6 {(0)}^{2} - 18 \cdot 0$ and solve for $x$ .

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Step 1.3.2.1

Remove parentheses.

$x = 6 \cdot 0^{2} - 18 \cdot 0$

Step 1.3.2.2

Remove parentheses.

$x = 6 {(0)}^{2} - 18 \cdot 0$

Step 1.3.2.3

Simplify $6 {(0)}^{2} - 18 \cdot 0$ .

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Step 1.3.2.3.1

Simplify each term.

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Step 1.3.2.3.1.1

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

$x = 6 \cdot 0 - 18 \cdot 0$

Step 1.3.2.3.1.2

Multiply $6$ by $0$ .

$x = 0 - 18 \cdot 0$

Step 1.3.2.3.1.3

Multiply $- 18$ by $0$ .

$x = 0 + 0$

Step 1.3.2.3.2

Add $0$ and $0$ .

$x = 0$

Step 1.4

Evaluate $x$ when $y = - \frac{3}{4}$ .

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Step 1.4.1

Substitute $- \frac{3}{4}$ for $y$ .

$x = 6 {(- \frac{3}{4})}^{2} - 18 (- \frac{3}{4})$

Step 1.4.2

Simplify $6 {(- \frac{3}{4})}^{2} - 18 (- \frac{3}{4})$ .

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Step 1.4.2.1

Simplify each term.

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Step 1.4.2.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 1.4.2.1.1.1

Apply the product rule to $- \frac{3}{4}$ .

$x = 6 ({(- 1)}^{2} {(\frac{3}{4})}^{2}) - 18 (- \frac{3}{4})$

Step 1.4.2.1.1.2

Apply the product rule to $\frac{3}{4}$ .

$x = 6 ({(- 1)}^{2} \frac{3^{2}}{4^{2}}) - 18 (- \frac{3}{4})$

Step 1.4.2.1.2

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

$x = 6 (1 \frac{3^{2}}{4^{2}}) - 18 (- \frac{3}{4})$

Step 1.4.2.1.3

Multiply $\frac{3^{2}}{4^{2}}$ by $1$ .

$x = 6 \frac{3^{2}}{4^{2}} - 18 (- \frac{3}{4})$

Step 1.4.2.1.4

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

$x = 6 \frac{9}{4^{2}} - 18 (- \frac{3}{4})$

Step 1.4.2.1.5

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

$x = 6 (\frac{9}{16}) - 18 (- \frac{3}{4})$

Step 1.4.2.1.6

Cancel the common factor of $2$ .

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Step 1.4.2.1.6.1

Factor $2$ out of $6$ .

$x = 2 (3) \frac{9}{16} - 18 (- \frac{3}{4})$

Step 1.4.2.1.6.2

Factor $2$ out of $16$ .

$x = 2 \cdot 3 \frac{9}{2 \cdot 8} - 18 (- \frac{3}{4})$

Step 1.4.2.1.6.3

Cancel the common factor.

$x = 2 \cdot 3 \frac{9}{2 \cdot 8} - 18 (- \frac{3}{4})$

Step 1.4.2.1.6.4

Rewrite the expression.

$x = 3 (\frac{9}{8}) - 18 (- \frac{3}{4})$

Step 1.4.2.1.7

Combine $3$ and $\frac{9}{8}$ .

$x = \frac{3 \cdot 9}{8} - 18 (- \frac{3}{4})$

Step 1.4.2.1.8

Multiply $3$ by $9$ .

$x = \frac{27}{8} - 18 (- \frac{3}{4})$

Step 1.4.2.1.9

Cancel the common factor of $2$ .

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Step 1.4.2.1.9.1

Move the leading negative in $- \frac{3}{4}$ into the numerator.

$x = \frac{27}{8} - 18 (\frac{- 3}{4})$

Step 1.4.2.1.9.2

Factor $2$ out of $- 18$ .

$x = \frac{27}{8} + 2 (- 9) \frac{- 3}{4}$

Step 1.4.2.1.9.3

Factor $2$ out of $4$ .

$x = \frac{27}{8} + 2 \cdot - 9 \frac{- 3}{2 \cdot 2}$

Step 1.4.2.1.9.4

Cancel the common factor.

$x = \frac{27}{8} + 2 \cdot - 9 \frac{- 3}{2 \cdot 2}$

Step 1.4.2.1.9.5

Rewrite the expression.

$x = \frac{27}{8} - 9 (\frac{- 3}{2})$

Step 1.4.2.1.10

Combine $- 9$ and $\frac{- 3}{2}$ .

$x = \frac{27}{8} + \frac{- 9 \cdot - 3}{2}$

Step 1.4.2.1.11

Multiply $- 9$ by $- 3$ .

$x = \frac{27}{8} + \frac{27}{2}$

Step 1.4.2.2

To write $\frac{27}{2}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = \frac{27}{8} + \frac{27}{2} \cdot \frac{4}{4}$

Step 1.4.2.3

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.4.2.3.1

Multiply $\frac{27}{2}$ by $\frac{4}{4}$ .

$x = \frac{27}{8} + \frac{27 \cdot 4}{2 \cdot 4}$

Step 1.4.2.3.2

Multiply $2$ by $4$ .

$x = \frac{27}{8} + \frac{27 \cdot 4}{8}$

Step 1.4.2.4

Combine the numerators over the common denominator.

$x = \frac{27 + 27 \cdot 4}{8}$

Step 1.4.2.5

Simplify the numerator.

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Step 1.4.2.5.1

Multiply $27$ by $4$ .

$x = \frac{27 + 108}{8}$

Step 1.4.2.5.2

Add $27$ and $108$ .

$x = \frac{135}{8}$

Step 1.5

Evaluate $x$ when $y = 3$ .

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Step 1.5.1

Substitute $3$ for $y$ .

$x = 6 {(3)}^{2} - 18 \cdot 3$

Step 1.5.2

Substitute $3$ for $y$ in $x = 6 {(3)}^{2} - 18 \cdot 3$ and solve for $x$ .

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Step 1.5.2.1

Remove parentheses.

$x = 6 \cdot 3^{2} - 18 \cdot 3$

Step 1.5.2.2

Remove parentheses.

$x = 6 {(3)}^{2} - 18 \cdot 3$

Step 1.5.2.3

Simplify $6 {(3)}^{2} - 18 \cdot 3$ .

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Step 1.5.2.3.1

Simplify each term.

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Step 1.5.2.3.1.1

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

$x = 6 \cdot 9 - 18 \cdot 3$

Step 1.5.2.3.1.2

Multiply $6$ by $9$ .

$x = 54 - 18 \cdot 3$

Step 1.5.2.3.1.3

Multiply $- 18$ by $3$ .

$x = 54 - 54$

Step 1.5.2.3.2

Subtract $54$ from $54$ .

$x = 0$

Step 1.6

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

$(0, 0)$

$(\frac{135}{8}, - \frac{3}{4})$

$(0, 3)$

$(0, 0)$

$(\frac{135}{8}, - \frac{3}{4})$

$(0, 3)$

Step 2

Reorder $24 y^{2}$ and $- 8 y^{3}$ .

$x = - 8 y^{3} + 24 y^{2}$

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.

$A r e a = \int_{- \frac{3}{4}}^{0} 6 y^{2} - 18 y d y - \int_{- \frac{3}{4}}^{0} - 8 y^{3} + 24 y^{2} d y$

Step 4

Integrate to find the area between $- \frac{3}{4}$ and $0$ .

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Step 4.1

Combine the integrals into a single integral.

$\int_{- \frac{3}{4}}^{0} 6 y^{2} - 18 y - (- 8 y^{3} + 24 y^{2}) d y$

Step 4.2

Simplify each term.

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Step 4.2.1

Apply the distributive property.

$6 y^{2} - 18 y - (- 8 y^{3}) - (24 y^{2})$

Step 4.2.2

Multiply $- 8$ by $- 1$ .

$6 y^{2} - 18 y + 8 y^{3} - (24 y^{2})$

Step 4.2.3

Multiply $24$ by $- 1$ .

$6 y^{2} - 18 y + 8 y^{3} - 24 y^{2}$

$\int_{- \frac{3}{4}}^{0} 6 y^{2} - 18 y + 8 y^{3} - 24 y^{2} d y$

Step 4.3

Subtract $24 y^{2}$ from $6 y^{2}$ .

$\int_{- \frac{3}{4}}^{0} - 18 y^{2} - 18 y + 8 y^{3} d y$

Step 4.4

Split the single integral into multiple integrals.

$\int_{- \frac{3}{4}}^{0} - 18 y^{2} d y + \int_{- \frac{3}{4}}^{0} - 18 y d y + \int_{- \frac{3}{4}}^{0} 8 y^{3} d y$

Step 4.5

Since $- 18$ is constant with respect to $y$ , move $- 18$ out of the integral.

$- 18 \int_{- \frac{3}{4}}^{0} y^{2} d y + \int_{- \frac{3}{4}}^{0} - 18 y d y + \int_{- \frac{3}{4}}^{0} 8 y^{3} d y$

Step 4.6

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

$- 18 (\frac{1}{3} y^{3}]_{- \frac{3}{4}}^{0}) + \int_{- \frac{3}{4}}^{0} - 18 y d y + \int_{- \frac{3}{4}}^{0} 8 y^{3} d y$

Step 4.7

Combine $\frac{1}{3}$ and $y^{3}$ .

$- 18 (\frac{y^{3}}{3}]_{- \frac{3}{4}}^{0}) + \int_{- \frac{3}{4}}^{0} - 18 y d y + \int_{- \frac{3}{4}}^{0} 8 y^{3} d y$

Step 4.8

Since $- 18$ is constant with respect to $y$ , move $- 18$ out of the integral.

$- 18 (\frac{y^{3}}{3}]_{- \frac{3}{4}}^{0}) - 18 \int_{- \frac{3}{4}}^{0} y d y + \int_{- \frac{3}{4}}^{0} 8 y^{3} d y$

Step 4.9

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

$- 18 (\frac{y^{3}}{3}]_{- \frac{3}{4}}^{0}) - 18 (\frac{1}{2} y^{2}]_{- \frac{3}{4}}^{0}) + \int_{- \frac{3}{4}}^{0} 8 y^{3} d y$

Step 4.10

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

$- 18 (\frac{y^{3}}{3}]_{- \frac{3}{4}}^{0}) - 18 (\frac{y^{2}}{2}]_{- \frac{3}{4}}^{0}) + \int_{- \frac{3}{4}}^{0} 8 y^{3} d y$

Step 4.11

Since $8$ is constant with respect to $y$ , move $8$ out of the integral.

$- 18 (\frac{y^{3}}{3}]_{- \frac{3}{4}}^{0}) - 18 (\frac{y^{2}}{2}]_{- \frac{3}{4}}^{0}) + 8 \int_{- \frac{3}{4}}^{0} y^{3} d y$

Step 4.12

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

$- 18 (\frac{y^{3}}{3}]_{- \frac{3}{4}}^{0}) - 18 (\frac{y^{2}}{2}]_{- \frac{3}{4}}^{0}) + 8 (\frac{1}{4} y^{4}]_{- \frac{3}{4}}^{0})$

Step 4.13

Simplify the answer.

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Step 4.13.1

Combine $\frac{1}{4}$ and $y^{4}$ .

$- 18 (\frac{y^{3}}{3}]_{- \frac{3}{4}}^{0}) - 18 (\frac{y^{2}}{2}]_{- \frac{3}{4}}^{0}) + 8 (\frac{y^{4}}{4}]_{- \frac{3}{4}}^{0})$

Step 4.13.2

Substitute and simplify.

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Step 4.13.2.1

Evaluate $\frac{y^{3}}{3}$ at $0$ and at $- \frac{3}{4}$ .

$- 18 ((\frac{0^{3}}{3}) - \frac{{(- \frac{3}{4})}^{3}}{3}) - 18 (\frac{y^{2}}{2}]_{- \frac{3}{4}}^{0}) + 8 (\frac{y^{4}}{4}]_{- \frac{3}{4}}^{0})$

Step 4.13.2.2

Evaluate $\frac{y^{2}}{2}$ at $0$ and at $- \frac{3}{4}$ .

$- 18 (\frac{0^{3}}{3} - \frac{{(- \frac{3}{4})}^{3}}{3}) - 18 (\frac{0^{2}}{2} - \frac{{(- \frac{3}{4})}^{2}}{2}) + 8 (\frac{y^{4}}{4}]_{- \frac{3}{4}}^{0})$

Step 4.13.2.3

Evaluate $\frac{y^{4}}{4}$ at $0$ and at $- \frac{3}{4}$ .

$- 18 (\frac{0^{3}}{3} - \frac{{(- \frac{3}{4})}^{3}}{3}) - 18 (\frac{0^{2}}{2} - \frac{{(- \frac{3}{4})}^{2}}{2}) + 8 ((\frac{0^{4}}{4}) - \frac{{(- \frac{3}{4})}^{4}}{4})$

Step 4.13.2.4

Simplify.

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Step 4.13.2.4.1

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

$- 18 (\frac{0}{3} - \frac{{(- \frac{3}{4})}^{3}}{3}) - 18 (\frac{0^{2}}{2} - \frac{{(- \frac{3}{4})}^{2}}{2}) + 8 ((\frac{0^{4}}{4}) - \frac{{(- \frac{3}{4})}^{4}}{4})$

Step 4.13.2.4.2

Cancel the common factor of $0$ and $3$ .

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Step 4.13.2.4.2.1

Factor $3$ out of $0$ .

$- 18 (\frac{3 (0)}{3} - \frac{{(- \frac{3}{4})}^{3}}{3}) - 18 (\frac{0^{2}}{2} - \frac{{(- \frac{3}{4})}^{2}}{2}) + 8 ((\frac{0^{4}}{4}) - \frac{{(- \frac{3}{4})}^{4}}{4})$

Step 4.13.2.4.2.2

Cancel the common factors.

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Step 4.13.2.4.2.2.1

Factor $3$ out of $3$ .

$- 18 (\frac{3 \cdot 0}{3 \cdot 1} - \frac{{(- \frac{3}{4})}^{3}}{3}) - 18 (\frac{0^{2}}{2} - \frac{{(- \frac{3}{4})}^{2}}{2}) + 8 ((\frac{0^{4}}{4}) - \frac{{(- \frac{3}{4})}^{4}}{4})$

Step 4.13.2.4.2.2.2

Cancel the common factor.

$- 18 (\frac{3 \cdot 0}{3 \cdot 1} - \frac{{(- \frac{3}{4})}^{3}}{3}) - 18 (\frac{0^{2}}{2} - \frac{{(- \frac{3}{4})}^{2}}{2}) + 8 ((\frac{0^{4}}{4}) - \frac{{(- \frac{3}{4})}^{4}}{4})$

Step 4.13.2.4.2.2.3

Rewrite the expression.

$- 18 (\frac{0}{1} - \frac{{(- \frac{3}{4})}^{3}}{3}) - 18 (\frac{0^{2}}{2} - \frac{{(- \frac{3}{4})}^{2}}{2}) + 8 ((\frac{0^{4}}{4}) - \frac{{(- \frac{3}{4})}^{4}}{4})$

Step 4.13.2.4.2.2.4

Divide $0$ by $1$ .

$- 18 (0 - \frac{{(- \frac{3}{4})}^{3}}{3}) - 18 (\frac{0^{2}}{2} - \frac{{(- \frac{3}{4})}^{2}}{2}) + 8 ((\frac{0^{4}}{4}) - \frac{{(- \frac{3}{4})}^{4}}{4})$

Step 4.13.2.4.3

Factor $- 1$ out of $- \frac{3}{4}$ .

$- 18 (0 - \frac{{(- (\frac{3}{4}))}^{3}}{3}) - 18 (\frac{0^{2}}{2} - \frac{{(- \frac{3}{4})}^{2}}{2}) + 8 ((\frac{0^{4}}{4}) - \frac{{(- \frac{3}{4})}^{4}}{4})$

Step 4.13.2.4.4

Apply the product rule to $- (\frac{3}{4})$ .

$- 18 (0 - \frac{{(- 1)}^{3} {(\frac{3}{4})}^{3}}{3}) - 18 (\frac{0^{2}}{2} - \frac{{(- \frac{3}{4})}^{2}}{2}) + 8 ((\frac{0^{4}}{4}) - \frac{{(- \frac{3}{4})}^{4}}{4})$

Step 4.13.2.4.5

Raise $- 1$ to the power of $3$ .

$- 18 (0 - \frac{- {(\frac{3}{4})}^{3}}{3}) - 18 (\frac{0^{2}}{2} - \frac{{(- \frac{3}{4})}^{2}}{2}) + 8 ((\frac{0^{4}}{4}) - \frac{{(- \frac{3}{4})}^{4}}{4})$

Step 4.13.2.4.6

Move the negative in front of the fraction.

$- 18 (0 - - \frac{{(\frac{3}{4})}^{3}}{3}) - 18 (\frac{0^{2}}{2} - \frac{{(- \frac{3}{4})}^{2}}{2}) + 8 ((\frac{0^{4}}{4}) - \frac{{(- \frac{3}{4})}^{4}}{4})$

Step 4.13.2.4.7

Multiply $- 1$ by $- 1$ .

$- 18 (0 + 1 \frac{{(\frac{3}{4})}^{3}}{3}) - 18 (\frac{0^{2}}{2} - \frac{{(- \frac{3}{4})}^{2}}{2}) + 8 ((\frac{0^{4}}{4}) - \frac{{(- \frac{3}{4})}^{4}}{4})$

Step 4.13.2.4.8

Multiply $\frac{{(\frac{3}{4})}^{3}}{3}$ by $1$ .

$- 18 (0 + \frac{{(\frac{3}{4})}^{3}}{3}) - 18 (\frac{0^{2}}{2} - \frac{{(- \frac{3}{4})}^{2}}{2}) + 8 ((\frac{0^{4}}{4}) - \frac{{(- \frac{3}{4})}^{4}}{4})$

Step 4.13.2.4.9

Add $0$ and $\frac{{(\frac{3}{4})}^{3}}{3}$ .

$- 18 \frac{{(\frac{3}{4})}^{3}}{3} - 18 (\frac{0^{2}}{2} - \frac{{(- \frac{3}{4})}^{2}}{2}) + 8 ((\frac{0^{4}}{4}) - \frac{{(- \frac{3}{4})}^{4}}{4})$

Step 4.13.2.4.10

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

$- 18 \frac{{(\frac{3}{4})}^{3}}{3} - 18 (\frac{0}{2} - \frac{{(- \frac{3}{4})}^{2}}{2}) + 8 ((\frac{0^{4}}{4}) - \frac{{(- \frac{3}{4})}^{4}}{4})$

Step 4.13.2.4.11

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

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Step 4.13.2.4.11.1

Factor $2$ out of $0$ .

$- 18 \frac{{(\frac{3}{4})}^{3}}{3} - 18 (\frac{2 (0)}{2} - \frac{{(- \frac{3}{4})}^{2}}{2}) + 8 ((\frac{0^{4}}{4}) - \frac{{(- \frac{3}{4})}^{4}}{4})$

Step 4.13.2.4.11.2

Cancel the common factors.

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Step 4.13.2.4.11.2.1

Factor $2$ out of $2$ .

$- 18 \frac{{(\frac{3}{4})}^{3}}{3} - 18 (\frac{2 \cdot 0}{2 \cdot 1} - \frac{{(- \frac{3}{4})}^{2}}{2}) + 8 ((\frac{0^{4}}{4}) - \frac{{(- \frac{3}{4})}^{4}}{4})$

Step 4.13.2.4.11.2.2

Cancel the common factor.

$- 18 \frac{{(\frac{3}{4})}^{3}}{3} - 18 (\frac{2 \cdot 0}{2 \cdot 1} - \frac{{(- \frac{3}{4})}^{2}}{2}) + 8 ((\frac{0^{4}}{4}) - \frac{{(- \frac{3}{4})}^{4}}{4})$

Step 4.13.2.4.11.2.3

Rewrite the expression.

$- 18 \frac{{(\frac{3}{4})}^{3}}{3} - 18 (\frac{0}{1} - \frac{{(- \frac{3}{4})}^{2}}{2}) + 8 ((\frac{0^{4}}{4}) - \frac{{(- \frac{3}{4})}^{4}}{4})$

Step 4.13.2.4.11.2.4

Divide $0$ by $1$ .

$- 18 \frac{{(\frac{3}{4})}^{3}}{3} - 18 (0 - \frac{{(- \frac{3}{4})}^{2}}{2}) + 8 ((\frac{0^{4}}{4}) - \frac{{(- \frac{3}{4})}^{4}}{4})$

Step 4.13.2.4.12

Factor $- 1$ out of $- \frac{3}{4}$ .

$- 18 \frac{{(\frac{3}{4})}^{3}}{3} - 18 (0 - \frac{{(- (\frac{3}{4}))}^{2}}{2}) + 8 ((\frac{0^{4}}{4}) - \frac{{(- \frac{3}{4})}^{4}}{4})$

Step 4.13.2.4.13

Apply the product rule to $- (\frac{3}{4})$ .

$- 18 \frac{{(\frac{3}{4})}^{3}}{3} - 18 (0 - \frac{{(- 1)}^{2} {(\frac{3}{4})}^{2}}{2}) + 8 ((\frac{0^{4}}{4}) - \frac{{(- \frac{3}{4})}^{4}}{4})$

Step 4.13.2.4.14

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

$- 18 \frac{{(\frac{3}{4})}^{3}}{3} - 18 (0 - \frac{1 {(\frac{3}{4})}^{2}}{2}) + 8 ((\frac{0^{4}}{4}) - \frac{{(- \frac{3}{4})}^{4}}{4})$

Step 4.13.2.4.15

Multiply ${(\frac{3}{4})}^{2}$ by $1$ .

$- 18 \frac{{(\frac{3}{4})}^{3}}{3} - 18 (0 - \frac{{(\frac{3}{4})}^{2}}{2}) + 8 ((\frac{0^{4}}{4}) - \frac{{(- \frac{3}{4})}^{4}}{4})$

Step 4.13.2.4.16

Subtract $\frac{{(\frac{3}{4})}^{2}}{2}$ from $0$ .

$- 18 \frac{{(\frac{3}{4})}^{3}}{3} - 18 (- \frac{{(\frac{3}{4})}^{2}}{2}) + 8 ((\frac{0^{4}}{4}) - \frac{{(- \frac{3}{4})}^{4}}{4})$

Step 4.13.2.4.17

Multiply $- 1$ by $- 18$ .

$- 18 \frac{{(\frac{3}{4})}^{3}}{3} + 18 \frac{{(\frac{3}{4})}^{2}}{2} + 8 ((\frac{0^{4}}{4}) - \frac{{(- \frac{3}{4})}^{4}}{4})$

Step 4.13.2.4.18

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

$- 18 \frac{{(\frac{3}{4})}^{3}}{3} + 18 \frac{{(\frac{3}{4})}^{2}}{2} + 8 (\frac{0}{4} - \frac{{(- \frac{3}{4})}^{4}}{4})$

Step 4.13.2.4.19

Cancel the common factor of $0$ and $4$ .

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Step 4.13.2.4.19.1

Factor $4$ out of $0$ .

$- 18 \frac{{(\frac{3}{4})}^{3}}{3} + 18 \frac{{(\frac{3}{4})}^{2}}{2} + 8 (\frac{4 (0)}{4} - \frac{{(- \frac{3}{4})}^{4}}{4})$

Step 4.13.2.4.19.2

Cancel the common factors.

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Step 4.13.2.4.19.2.1

Factor $4$ out of $4$ .

$- 18 \frac{{(\frac{3}{4})}^{3}}{3} + 18 \frac{{(\frac{3}{4})}^{2}}{2} + 8 (\frac{4 \cdot 0}{4 \cdot 1} - \frac{{(- \frac{3}{4})}^{4}}{4})$

Step 4.13.2.4.19.2.2

Cancel the common factor.

$- 18 \frac{{(\frac{3}{4})}^{3}}{3} + 18 \frac{{(\frac{3}{4})}^{2}}{2} + 8 (\frac{4 \cdot 0}{4 \cdot 1} - \frac{{(- \frac{3}{4})}^{4}}{4})$

Step 4.13.2.4.19.2.3

Rewrite the expression.

$- 18 \frac{{(\frac{3}{4})}^{3}}{3} + 18 \frac{{(\frac{3}{4})}^{2}}{2} + 8 (\frac{0}{1} - \frac{{(- \frac{3}{4})}^{4}}{4})$

Step 4.13.2.4.19.2.4

Divide $0$ by $1$ .

$- 18 \frac{{(\frac{3}{4})}^{3}}{3} + 18 \frac{{(\frac{3}{4})}^{2}}{2} + 8 (0 - \frac{{(- \frac{3}{4})}^{4}}{4})$

Step 4.13.2.4.20

Factor $- 1$ out of $- \frac{3}{4}$ .

$- 18 \frac{{(\frac{3}{4})}^{3}}{3} + 18 \frac{{(\frac{3}{4})}^{2}}{2} + 8 (0 - \frac{{(- (\frac{3}{4}))}^{4}}{4})$

Step 4.13.2.4.21

Apply the product rule to $- (\frac{3}{4})$ .

$- 18 \frac{{(\frac{3}{4})}^{3}}{3} + 18 \frac{{(\frac{3}{4})}^{2}}{2} + 8 (0 - \frac{{(- 1)}^{4} {(\frac{3}{4})}^{4}}{4})$

Step 4.13.2.4.22

Raise $- 1$ to the power of $4$ .

$- 18 \frac{{(\frac{3}{4})}^{3}}{3} + 18 \frac{{(\frac{3}{4})}^{2}}{2} + 8 (0 - \frac{1 {(\frac{3}{4})}^{4}}{4})$

Step 4.13.2.4.23

Multiply ${(\frac{3}{4})}^{4}$ by $1$ .

$- 18 \frac{{(\frac{3}{4})}^{3}}{3} + 18 \frac{{(\frac{3}{4})}^{2}}{2} + 8 (0 - \frac{{(\frac{3}{4})}^{4}}{4})$

Step 4.13.2.4.24

Subtract $\frac{{(\frac{3}{4})}^{4}}{4}$ from $0$ .

$- 18 \frac{{(\frac{3}{4})}^{3}}{3} + 18 \frac{{(\frac{3}{4})}^{2}}{2} + 8 (- \frac{{(\frac{3}{4})}^{4}}{4})$

Step 4.13.2.4.25

Multiply $- 1$ by $8$ .

$- 18 \frac{{(\frac{3}{4})}^{3}}{3} + 18 \frac{{(\frac{3}{4})}^{2}}{2} - 8 \frac{{(\frac{3}{4})}^{4}}{4}$

Step 4.13.3

Simplify.

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Step 4.13.3.1

Simplify each term.

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Step 4.13.3.1.1

Cancel the common factor of $3$ .

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Step 4.13.3.1.1.1

Factor $3$ out of $- 18$ .

$3 (- 6) \frac{{(\frac{3}{4})}^{3}}{3} + 18 \frac{{(\frac{3}{4})}^{2}}{2} - 8 \frac{{(\frac{3}{4})}^{4}}{4}$

Step 4.13.3.1.1.2

Cancel the common factor.

$3 \cdot - 6 \frac{{(\frac{3}{4})}^{3}}{3} + 18 \frac{{(\frac{3}{4})}^{2}}{2} - 8 \frac{{(\frac{3}{4})}^{4}}{4}$

Step 4.13.3.1.1.3

Rewrite the expression.

$- 6 {(\frac{3}{4})}^{3} + 18 \frac{{(\frac{3}{4})}^{2}}{2} - 8 \frac{{(\frac{3}{4})}^{4}}{4}$

Step 4.13.3.1.2

Apply the product rule to $\frac{3}{4}$ .

$- 6 \frac{3^{3}}{4^{3}} + 18 \frac{{(\frac{3}{4})}^{2}}{2} - 8 \frac{{(\frac{3}{4})}^{4}}{4}$

Step 4.13.3.1.3

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

$- 6 \frac{27}{4^{3}} + 18 \frac{{(\frac{3}{4})}^{2}}{2} - 8 \frac{{(\frac{3}{4})}^{4}}{4}$

Step 4.13.3.1.4

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

$- 6 (\frac{27}{64}) + 18 \frac{{(\frac{3}{4})}^{2}}{2} - 8 \frac{{(\frac{3}{4})}^{4}}{4}$

Step 4.13.3.1.5

Cancel the common factor of $2$ .

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Step 4.13.3.1.5.1

Factor $2$ out of $- 6$ .

$2 (- 3) \frac{27}{64} + 18 \frac{{(\frac{3}{4})}^{2}}{2} - 8 \frac{{(\frac{3}{4})}^{4}}{4}$

Step 4.13.3.1.5.2

Factor $2$ out of $64$ .

$2 \cdot - 3 \frac{27}{2 \cdot 32} + 18 \frac{{(\frac{3}{4})}^{2}}{2} - 8 \frac{{(\frac{3}{4})}^{4}}{4}$

Step 4.13.3.1.5.3

Cancel the common factor.

$2 \cdot - 3 \frac{27}{2 \cdot 32} + 18 \frac{{(\frac{3}{4})}^{2}}{2} - 8 \frac{{(\frac{3}{4})}^{4}}{4}$

Step 4.13.3.1.5.4

Rewrite the expression.

$- 3 (\frac{27}{32}) + 18 \frac{{(\frac{3}{4})}^{2}}{2} - 8 \frac{{(\frac{3}{4})}^{4}}{4}$

Step 4.13.3.1.6

Combine $- 3$ and $\frac{27}{32}$ .

$\frac{- 3 \cdot 27}{32} + 18 \frac{{(\frac{3}{4})}^{2}}{2} - 8 \frac{{(\frac{3}{4})}^{4}}{4}$

Step 4.13.3.1.7

Multiply $- 3$ by $27$ .

$\frac{- 81}{32} + 18 \frac{{(\frac{3}{4})}^{2}}{2} - 8 \frac{{(\frac{3}{4})}^{4}}{4}$

Step 4.13.3.1.8

Move the negative in front of the fraction.

$- \frac{81}{32} + 18 \frac{{(\frac{3}{4})}^{2}}{2} - 8 \frac{{(\frac{3}{4})}^{4}}{4}$

Step 4.13.3.1.9

Cancel the common factor of $2$ .

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Step 4.13.3.1.9.1

Factor $2$ out of $18$ .

$- \frac{81}{32} + 2 (9) \frac{{(\frac{3}{4})}^{2}}{2} - 8 \frac{{(\frac{3}{4})}^{4}}{4}$

Step 4.13.3.1.9.2

Cancel the common factor.

$- \frac{81}{32} + 2 \cdot 9 \frac{{(\frac{3}{4})}^{2}}{2} - 8 \frac{{(\frac{3}{4})}^{4}}{4}$

Step 4.13.3.1.9.3

Rewrite the expression.

$- \frac{81}{32} + 9 {(\frac{3}{4})}^{2} - 8 \frac{{(\frac{3}{4})}^{4}}{4}$

Step 4.13.3.1.10

Apply the product rule to $\frac{3}{4}$ .

$- \frac{81}{32} + 9 \frac{3^{2}}{4^{2}} - 8 \frac{{(\frac{3}{4})}^{4}}{4}$

Step 4.13.3.1.11

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

$- \frac{81}{32} + 9 \frac{9}{4^{2}} - 8 \frac{{(\frac{3}{4})}^{4}}{4}$

Step 4.13.3.1.12

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

$- \frac{81}{32} + 9 (\frac{9}{16}) - 8 \frac{{(\frac{3}{4})}^{4}}{4}$

Step 4.13.3.1.13

Multiply $9 (\frac{9}{16})$ .

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Step 4.13.3.1.13.1

Combine $9$ and $\frac{9}{16}$ .

$- \frac{81}{32} + \frac{9 \cdot 9}{16} - 8 \frac{{(\frac{3}{4})}^{4}}{4}$

Step 4.13.3.1.13.2

Multiply $9$ by $9$ .

$- \frac{81}{32} + \frac{81}{16} - 8 \frac{{(\frac{3}{4})}^{4}}{4}$

Step 4.13.3.1.14

Cancel the common factor of $4$ .

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Step 4.13.3.1.14.1

Factor $4$ out of $- 8$ .

$- \frac{81}{32} + \frac{81}{16} + 4 (- 2) \frac{{(\frac{3}{4})}^{4}}{4}$

Step 4.13.3.1.14.2

Cancel the common factor.

$- \frac{81}{32} + \frac{81}{16} + 4 \cdot - 2 \frac{{(\frac{3}{4})}^{4}}{4}$

Step 4.13.3.1.14.3

Rewrite the expression.

$- \frac{81}{32} + \frac{81}{16} - 2 {(\frac{3}{4})}^{4}$

Step 4.13.3.1.15

Apply the product rule to $\frac{3}{4}$ .

$- \frac{81}{32} + \frac{81}{16} - 2 \frac{3^{4}}{4^{4}}$

Step 4.13.3.1.16

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

$- \frac{81}{32} + \frac{81}{16} - 2 \frac{81}{4^{4}}$

Step 4.13.3.1.17

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

$- \frac{81}{32} + \frac{81}{16} - 2 (\frac{81}{256})$

Step 4.13.3.1.18

Cancel the common factor of $2$ .

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Step 4.13.3.1.18.1

Factor $2$ out of $- 2$ .

$- \frac{81}{32} + \frac{81}{16} + 2 (- 1) \frac{81}{256}$

Step 4.13.3.1.18.2

Factor $2$ out of $256$ .

$- \frac{81}{32} + \frac{81}{16} + 2 \cdot - 1 \frac{81}{2 \cdot 128}$

Step 4.13.3.1.18.3

Cancel the common factor.

$- \frac{81}{32} + \frac{81}{16} + 2 \cdot - 1 \frac{81}{2 \cdot 128}$

Step 4.13.3.1.18.4

Rewrite the expression.

$- \frac{81}{32} + \frac{81}{16} - \frac{81}{128}$

Step 4.13.3.2

Find the common denominator.

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Step 4.13.3.2.1

Multiply $\frac{81}{32}$ by $\frac{4}{4}$ .

$- (\frac{81}{32} \cdot \frac{4}{4}) + \frac{81}{16} - \frac{81}{128}$

Step 4.13.3.2.2

Multiply $\frac{81}{32}$ by $\frac{4}{4}$ .

$- \frac{81 \cdot 4}{32 \cdot 4} + \frac{81}{16} - \frac{81}{128}$

Step 4.13.3.2.3

Multiply $\frac{81}{16}$ by $\frac{8}{8}$ .

$- \frac{81 \cdot 4}{32 \cdot 4} + \frac{81}{16} \cdot \frac{8}{8} - \frac{81}{128}$

Step 4.13.3.2.4

Multiply $\frac{81}{16}$ by $\frac{8}{8}$ .

$- \frac{81 \cdot 4}{32 \cdot 4} + \frac{81 \cdot 8}{16 \cdot 8} - \frac{81}{128}$

Step 4.13.3.2.5

Reorder the factors of $32 \cdot 4$ .

$- \frac{81 \cdot 4}{4 \cdot 32} + \frac{81 \cdot 8}{16 \cdot 8} - \frac{81}{128}$

Step 4.13.3.2.6

Multiply $4$ by $32$ .

$- \frac{81 \cdot 4}{128} + \frac{81 \cdot 8}{16 \cdot 8} - \frac{81}{128}$

Step 4.13.3.2.7

Reorder the factors of $16 \cdot 8$ .

$- \frac{81 \cdot 4}{128} + \frac{81 \cdot 8}{8 \cdot 16} - \frac{81}{128}$

Step 4.13.3.2.8

Multiply $8$ by $16$ .

$- \frac{81 \cdot 4}{128} + \frac{81 \cdot 8}{128} - \frac{81}{128}$

Step 4.13.3.3

Combine the numerators over the common denominator.

$\frac{- 81 \cdot 4 + 81 \cdot 8 - 81}{128}$

Step 4.13.3.4

Simplify each term.

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Step 4.13.3.4.1

Multiply $- 81$ by $4$ .

$\frac{- 324 + 81 \cdot 8 - 81}{128}$

Step 4.13.3.4.2

Multiply $81$ by $8$ .

$\frac{- 324 + 648 - 81}{128}$

Step 4.13.3.5

Add $- 324$ and $648$ .

$\frac{324 - 81}{128}$

Step 4.13.3.6

Subtract $81$ from $324$ .

$\frac{243}{128}$

Step 5

$A r e a = \int_{0}^{3} - 8 y^{3} + 24 y^{2} d y - \int_{0}^{3} 6 y^{2} - 18 y d y$

Step 6

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

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Step 6.1

Combine the integrals into a single integral.

$\int_{0}^{3} - 8 y^{3} + 24 y^{2} - (6 y^{2} - 18 y) d y$

Step 6.2

Simplify each term.

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Step 6.2.1

Apply the distributive property.

$- 8 y^{3} + 24 y^{2} - (6 y^{2}) - (- 18 y)$

Step 6.2.2

Multiply $6$ by $- 1$ .

$- 8 y^{3} + 24 y^{2} - 6 y^{2} - (- 18 y)$

Step 6.2.3

Multiply $- 18$ by $- 1$ .

$- 8 y^{3} + 24 y^{2} - 6 y^{2} + 18 y$

$\int_{0}^{3} - 8 y^{3} + 24 y^{2} - 6 y^{2} + 18 y d y$

Step 6.3

Subtract $6 y^{2}$ from $24 y^{2}$ .

$\int_{0}^{3} - 8 y^{3} + 18 y^{2} + 18 y d y$

Step 6.4

Split the single integral into multiple integrals.

$\int_{0}^{3} - 8 y^{3} d y + \int_{0}^{3} 18 y^{2} d y + \int_{0}^{3} 18 y d y$

Step 6.5

Since $- 8$ is constant with respect to $y$ , move $- 8$ out of the integral.

$- 8 \int_{0}^{3} y^{3} d y + \int_{0}^{3} 18 y^{2} d y + \int_{0}^{3} 18 y d y$

Step 6.6

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

$- 8 (\frac{1}{4} y^{4}]_{0}^{3}) + \int_{0}^{3} 18 y^{2} d y + \int_{0}^{3} 18 y d y$

Step 6.7

Combine $\frac{1}{4}$ and $y^{4}$ .

$- 8 (\frac{y^{4}}{4}]_{0}^{3}) + \int_{0}^{3} 18 y^{2} d y + \int_{0}^{3} 18 y d y$

Step 6.8

Since $18$ is constant with respect to $y$ , move $18$ out of the integral.

$- 8 (\frac{y^{4}}{4}]_{0}^{3}) + 18 \int_{0}^{3} y^{2} d y + \int_{0}^{3} 18 y d y$

Step 6.9

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

$- 8 (\frac{y^{4}}{4}]_{0}^{3}) + 18 (\frac{1}{3} y^{3}]_{0}^{3}) + \int_{0}^{3} 18 y d y$

Step 6.10

Combine $\frac{1}{3}$ and $y^{3}$ .

$- 8 (\frac{y^{4}}{4}]_{0}^{3}) + 18 (\frac{y^{3}}{3}]_{0}^{3}) + \int_{0}^{3} 18 y d y$

Step 6.11

Since $18$ is constant with respect to $y$ , move $18$ out of the integral.

$- 8 (\frac{y^{4}}{4}]_{0}^{3}) + 18 (\frac{y^{3}}{3}]_{0}^{3}) + 18 \int_{0}^{3} y d y$

Step 6.12

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

$- 8 (\frac{y^{4}}{4}]_{0}^{3}) + 18 (\frac{y^{3}}{3}]_{0}^{3}) + 18 (\frac{1}{2} y^{2}]_{0}^{3})$

Step 6.13

Simplify the answer.

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Step 6.13.1

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

$- 8 (\frac{y^{4}}{4}]_{0}^{3}) + 18 (\frac{y^{3}}{3}]_{0}^{3}) + 18 (\frac{y^{2}}{2}]_{0}^{3})$

Step 6.13.2

Substitute and simplify.

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Step 6.13.2.1

Evaluate $\frac{y^{4}}{4}$ at $3$ and at $0$ .

$- 8 ((\frac{3^{4}}{4}) - \frac{0^{4}}{4}) + 18 (\frac{y^{3}}{3}]_{0}^{3}) + 18 (\frac{y^{2}}{2}]_{0}^{3})$

Step 6.13.2.2

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

$- 8 (\frac{3^{4}}{4} - \frac{0^{4}}{4}) + 18 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) + 18 (\frac{y^{2}}{2}]_{0}^{3})$

Step 6.13.2.3

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

$- 8 (\frac{3^{4}}{4} - \frac{0^{4}}{4}) + 18 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) + 18 ((\frac{3^{2}}{2}) - \frac{0^{2}}{2})$

Step 6.13.2.4

Simplify.

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Step 6.13.2.4.1

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

$- 8 (\frac{81}{4} - \frac{0^{4}}{4}) + 18 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) + 18 ((\frac{3^{2}}{2}) - \frac{0^{2}}{2})$

Step 6.13.2.4.2

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

$- 8 (\frac{81}{4} - \frac{0}{4}) + 18 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) + 18 ((\frac{3^{2}}{2}) - \frac{0^{2}}{2})$

Step 6.13.2.4.3

Cancel the common factor of $0$ and $4$ .

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Step 6.13.2.4.3.1

Factor $4$ out of $0$ .

$- 8 (\frac{81}{4} - \frac{4 (0)}{4}) + 18 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) + 18 ((\frac{3^{2}}{2}) - \frac{0^{2}}{2})$

Step 6.13.2.4.3.2

Cancel the common factors.

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Step 6.13.2.4.3.2.1

Factor $4$ out of $4$ .

$- 8 (\frac{81}{4} - \frac{4 \cdot 0}{4 \cdot 1}) + 18 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) + 18 ((\frac{3^{2}}{2}) - \frac{0^{2}}{2})$

Step 6.13.2.4.3.2.2

Cancel the common factor.

$- 8 (\frac{81}{4} - \frac{4 \cdot 0}{4 \cdot 1}) + 18 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) + 18 ((\frac{3^{2}}{2}) - \frac{0^{2}}{2})$

Step 6.13.2.4.3.2.3

Rewrite the expression.

$- 8 (\frac{81}{4} - \frac{0}{1}) + 18 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) + 18 ((\frac{3^{2}}{2}) - \frac{0^{2}}{2})$

Step 6.13.2.4.3.2.4

Divide $0$ by $1$ .

$- 8 (\frac{81}{4} - 0) + 18 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) + 18 ((\frac{3^{2}}{2}) - \frac{0^{2}}{2})$

Step 6.13.2.4.4

Multiply $- 1$ by $0$ .

$- 8 (\frac{81}{4} + 0) + 18 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) + 18 ((\frac{3^{2}}{2}) - \frac{0^{2}}{2})$

Step 6.13.2.4.5

Add $\frac{81}{4}$ and $0$ .

$- 8 (\frac{81}{4}) + 18 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) + 18 ((\frac{3^{2}}{2}) - \frac{0^{2}}{2})$

Step 6.13.2.4.6

Combine $- 8$ and $\frac{81}{4}$ .

$\frac{- 8 \cdot 81}{4} + 18 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) + 18 ((\frac{3^{2}}{2}) - \frac{0^{2}}{2})$

Step 6.13.2.4.7

Multiply $- 8$ by $81$ .

$\frac{- 648}{4} + 18 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) + 18 ((\frac{3^{2}}{2}) - \frac{0^{2}}{2})$

Step 6.13.2.4.8

Cancel the common factor of $- 648$ and $4$ .

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Step 6.13.2.4.8.1

Factor $4$ out of $- 648$ .

$\frac{4 \cdot - 162}{4} + 18 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) + 18 ((\frac{3^{2}}{2}) - \frac{0^{2}}{2})$

Step 6.13.2.4.8.2

Cancel the common factors.

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Step 6.13.2.4.8.2.1

Factor $4$ out of $4$ .

$\frac{4 \cdot - 162}{4 (1)} + 18 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) + 18 ((\frac{3^{2}}{2}) - \frac{0^{2}}{2})$

Step 6.13.2.4.8.2.2

Cancel the common factor.

$\frac{4 \cdot - 162}{4 \cdot 1} + 18 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) + 18 ((\frac{3^{2}}{2}) - \frac{0^{2}}{2})$

Step 6.13.2.4.8.2.3

Rewrite the expression.

$\frac{- 162}{1} + 18 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) + 18 ((\frac{3^{2}}{2}) - \frac{0^{2}}{2})$

Step 6.13.2.4.8.2.4

Divide $- 162$ by $1$ .

$- 162 + 18 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) + 18 ((\frac{3^{2}}{2}) - \frac{0^{2}}{2})$

Step 6.13.2.4.9

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

$- 162 + 18 (\frac{27}{3} - \frac{0^{3}}{3}) + 18 ((\frac{3^{2}}{2}) - \frac{0^{2}}{2})$

Step 6.13.2.4.10

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

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Step 6.13.2.4.10.1

Factor $3$ out of $27$ .

$- 162 + 18 (\frac{3 \cdot 9}{3} - \frac{0^{3}}{3}) + 18 ((\frac{3^{2}}{2}) - \frac{0^{2}}{2})$

Step 6.13.2.4.10.2

Cancel the common factors.

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Step 6.13.2.4.10.2.1

Factor $3$ out of $3$ .

$- 162 + 18 (\frac{3 \cdot 9}{3 (1)} - \frac{0^{3}}{3}) + 18 ((\frac{3^{2}}{2}) - \frac{0^{2}}{2})$

Step 6.13.2.4.10.2.2

Cancel the common factor.

$- 162 + 18 (\frac{3 \cdot 9}{3 \cdot 1} - \frac{0^{3}}{3}) + 18 ((\frac{3^{2}}{2}) - \frac{0^{2}}{2})$

Step 6.13.2.4.10.2.3

Rewrite the expression.

$- 162 + 18 (\frac{9}{1} - \frac{0^{3}}{3}) + 18 ((\frac{3^{2}}{2}) - \frac{0^{2}}{2})$

Step 6.13.2.4.10.2.4

Divide $9$ by $1$ .

$- 162 + 18 (9 - \frac{0^{3}}{3}) + 18 ((\frac{3^{2}}{2}) - \frac{0^{2}}{2})$

Step 6.13.2.4.11

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

$- 162 + 18 (9 - \frac{0}{3}) + 18 ((\frac{3^{2}}{2}) - \frac{0^{2}}{2})$

Step 6.13.2.4.12

Cancel the common factor of $0$ and $3$ .

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Step 6.13.2.4.12.1

Factor $3$ out of $0$ .

$- 162 + 18 (9 - \frac{3 (0)}{3}) + 18 ((\frac{3^{2}}{2}) - \frac{0^{2}}{2})$

Step 6.13.2.4.12.2

Cancel the common factors.

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Step 6.13.2.4.12.2.1

Factor $3$ out of $3$ .

$- 162 + 18 (9 - \frac{3 \cdot 0}{3 \cdot 1}) + 18 ((\frac{3^{2}}{2}) - \frac{0^{2}}{2})$

Step 6.13.2.4.12.2.2

Cancel the common factor.

$- 162 + 18 (9 - \frac{3 \cdot 0}{3 \cdot 1}) + 18 ((\frac{3^{2}}{2}) - \frac{0^{2}}{2})$

Step 6.13.2.4.12.2.3

Rewrite the expression.

$- 162 + 18 (9 - \frac{0}{1}) + 18 ((\frac{3^{2}}{2}) - \frac{0^{2}}{2})$

Step 6.13.2.4.12.2.4

Divide $0$ by $1$ .

$- 162 + 18 (9 - 0) + 18 ((\frac{3^{2}}{2}) - \frac{0^{2}}{2})$

Step 6.13.2.4.13

Multiply $- 1$ by $0$ .

$- 162 + 18 (9 + 0) + 18 ((\frac{3^{2}}{2}) - \frac{0^{2}}{2})$

Step 6.13.2.4.14

Add $9$ and $0$ .

$- 162 + 18 \cdot 9 + 18 ((\frac{3^{2}}{2}) - \frac{0^{2}}{2})$

Step 6.13.2.4.15

Multiply $18$ by $9$ .

$- 162 + 162 + 18 ((\frac{3^{2}}{2}) - \frac{0^{2}}{2})$

Step 6.13.2.4.16

Add $- 162$ and $162$ .

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

Step 6.13.2.4.17

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

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

Step 6.13.2.4.18

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

$0 + 18 (\frac{9}{2} - \frac{0}{2})$

Step 6.13.2.4.19

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

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Step 6.13.2.4.19.1

Factor $2$ out of $0$ .

$0 + 18 (\frac{9}{2} - \frac{2 (0)}{2})$

Step 6.13.2.4.19.2

Cancel the common factors.

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Step 6.13.2.4.19.2.1

Factor $2$ out of $2$ .

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

Step 6.13.2.4.19.2.2

Cancel the common factor.

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

Step 6.13.2.4.19.2.3

Rewrite the expression.

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

Step 6.13.2.4.19.2.4

Divide $0$ by $1$ .

$0 + 18 (\frac{9}{2} - 0)$

Step 6.13.2.4.20

Multiply $- 1$ by $0$ .

$0 + 18 (\frac{9}{2} + 0)$

Step 6.13.2.4.21

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

$0 + 18 (\frac{9}{2})$

Step 6.13.2.4.22

Combine $18$ and $\frac{9}{2}$ .

$0 + \frac{18 \cdot 9}{2}$

Step 6.13.2.4.23

Multiply $18$ by $9$ .

$0 + \frac{162}{2}$

Step 6.13.2.4.24

Cancel the common factor of $162$ and $2$ .

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Step 6.13.2.4.24.1

Factor $2$ out of $162$ .

$0 + \frac{2 \cdot 81}{2}$

Step 6.13.2.4.24.2

Cancel the common factors.

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Step 6.13.2.4.24.2.1

Factor $2$ out of $2$ .

$0 + \frac{2 \cdot 81}{2 (1)}$

Step 6.13.2.4.24.2.2

Cancel the common factor.

$0 + \frac{2 \cdot 81}{2 \cdot 1}$

Step 6.13.2.4.24.2.3

Rewrite the expression.

$0 + \frac{81}{1}$

Step 6.13.2.4.24.2.4

Divide $81$ by $1$ .

$0 + 81$

Step 6.13.2.4.25

Add $0$ and $81$ .

$81$

Step 7

Add the areas $\frac{243}{128} + 81$ .

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Step 7.1

To write $81$ as a fraction with a common denominator, multiply by $\frac{128}{128}$ .

$\frac{243}{128} + 81 \cdot \frac{128}{128}$

Step 7.2

Combine $81$ and $\frac{128}{128}$ .

$\frac{243}{128} + \frac{81 \cdot 128}{128}$

Step 7.3

Combine the numerators over the common denominator.

$\frac{243 + 81 \cdot 128}{128}$

Step 7.4

Simplify the numerator.

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Step 7.4.1

Multiply $81$ by $128$ .

$\frac{243 + 10368}{128}$

Step 7.4.2

Add $243$ and $10368$ .

$\frac{10611}{128}$

Step 8