Calculus Examples

$y = 3 + 2 x - x^{2}$ , $x + y = 3$

Step 1

To find the volume of the solid, first define the area of each slice then integrate across the range. The area of each slice is the area of a circle with radius $f (x)$ and $A = π r^{2}$ .

$V = π \int_{0}^{3} {(f (x))}^{2} - {(g (x))}^{2} d x$ where $f (x) = - x^{2} + 2 x + 3$ and $g (x) = 3 - x$

Step 2

Simplify the integrand.

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite ${(- x^{2} + 2 x + 3)}^{2}$ as $(- x^{2} + 2 x + 3) (- x^{2} + 2 x + 3)$ .

$V = (- x^{2} + 2 x + 3) (- x^{2} + 2 x + 3) - {(3 - x)}^{2}$

Step 2.1.2

Expand $(- x^{2} + 2 x + 3) (- x^{2} + 2 x + 3)$ by multiplying each term in the first expression by each term in the second expression.

$V = - x^{2} (- x^{2}) - x^{2} (2 x) - x^{2} \cdot 3 + 2 x (- x^{2}) + 2 x (2 x) + 2 x \cdot 3 + 3 (- x^{2}) + 3 (2 x) + 3 \cdot 3 - {(3 - x)}^{2}$

Step 2.1.3

Simplify each term.

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Step 2.1.3.1

Rewrite using the commutative property of multiplication.

$V = - 1 \cdot (- 1 x^{2} x^{2}) - x^{2} (2 x) - x^{2} \cdot 3 + 2 x (- x^{2}) + 2 x (2 x) + 2 x \cdot 3 + 3 (- x^{2}) + 3 (2 x) + 3 \cdot 3 - {(3 - x)}^{2}$

Step 2.1.3.2

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 2.1.3.2.1

Move $x^{2}$ .

$V = - 1 \cdot (- 1 (x^{2} x^{2})) - x^{2} (2 x) - x^{2} \cdot 3 + 2 x (- x^{2}) + 2 x (2 x) + 2 x \cdot 3 + 3 (- x^{2}) + 3 (2 x) + 3 \cdot 3 - {(3 - x)}^{2}$

Step 2.1.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$V = - 1 \cdot (- 1 x^{2 + 2}) - x^{2} (2 x) - x^{2} \cdot 3 + 2 x (- x^{2}) + 2 x (2 x) + 2 x \cdot 3 + 3 (- x^{2}) + 3 (2 x) + 3 \cdot 3 - {(3 - x)}^{2}$

Step 2.1.3.2.3

Add $2$ and $2$ .

$V = - 1 \cdot (- 1 x^{4}) - x^{2} (2 x) - x^{2} \cdot 3 + 2 x (- x^{2}) + 2 x (2 x) + 2 x \cdot 3 + 3 (- x^{2}) + 3 (2 x) + 3 \cdot 3 - {(3 - x)}^{2}$

Step 2.1.3.3

Multiply $- 1$ by $- 1$ .

$V = 1 x^{4} - x^{2} (2 x) - x^{2} \cdot 3 + 2 x (- x^{2}) + 2 x (2 x) + 2 x \cdot 3 + 3 (- x^{2}) + 3 (2 x) + 3 \cdot 3 - {(3 - x)}^{2}$

Step 2.1.3.4

Multiply $x^{4}$ by $1$ .

$V = x^{4} - x^{2} (2 x) - x^{2} \cdot 3 + 2 x (- x^{2}) + 2 x (2 x) + 2 x \cdot 3 + 3 (- x^{2}) + 3 (2 x) + 3 \cdot 3 - {(3 - x)}^{2}$

Step 2.1.3.5

Rewrite using the commutative property of multiplication.

$V = x^{4} - 1 \cdot (2 x^{2} x) - x^{2} \cdot 3 + 2 x (- x^{2}) + 2 x (2 x) + 2 x \cdot 3 + 3 (- x^{2}) + 3 (2 x) + 3 \cdot 3 - {(3 - x)}^{2}$

Step 2.1.3.6

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 2.1.3.6.1

Move $x$ .

$V = x^{4} - 1 \cdot (2 (x \cdot x^{2})) - x^{2} \cdot 3 + 2 x (- x^{2}) + 2 x (2 x) + 2 x \cdot 3 + 3 (- x^{2}) + 3 (2 x) + 3 \cdot 3 - {(3 - x)}^{2}$

Step 2.1.3.6.2

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

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Step 2.1.3.6.2.1

Raise $x$ to the power of $1$ .

$V = x^{4} - 1 \cdot (2 (x \cdot x^{2})) - x^{2} \cdot 3 + 2 x (- x^{2}) + 2 x (2 x) + 2 x \cdot 3 + 3 (- x^{2}) + 3 (2 x) + 3 \cdot 3 - {(3 - x)}^{2}$

Step 2.1.3.6.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$V = x^{4} - 1 \cdot (2 x^{1 + 2}) - x^{2} \cdot 3 + 2 x (- x^{2}) + 2 x (2 x) + 2 x \cdot 3 + 3 (- x^{2}) + 3 (2 x) + 3 \cdot 3 - {(3 - x)}^{2}$

Step 2.1.3.6.3

Add $1$ and $2$ .

$V = x^{4} - 1 \cdot (2 x^{3}) - x^{2} \cdot 3 + 2 x (- x^{2}) + 2 x (2 x) + 2 x \cdot 3 + 3 (- x^{2}) + 3 (2 x) + 3 \cdot 3 - {(3 - x)}^{2}$

Step 2.1.3.7

Multiply $- 1$ by $2$ .

$V = x^{4} - 2 x^{3} - x^{2} \cdot 3 + 2 x (- x^{2}) + 2 x (2 x) + 2 x \cdot 3 + 3 (- x^{2}) + 3 (2 x) + 3 \cdot 3 - {(3 - x)}^{2}$

Step 2.1.3.8

Multiply $3$ by $- 1$ .

$V = x^{4} - 2 x^{3} - 3 x^{2} + 2 x (- x^{2}) + 2 x (2 x) + 2 x \cdot 3 + 3 (- x^{2}) + 3 (2 x) + 3 \cdot 3 - {(3 - x)}^{2}$

Step 2.1.3.9

Rewrite using the commutative property of multiplication.

$V = x^{4} - 2 x^{3} - 3 x^{2} + 2 \cdot (- 1 x \cdot x^{2}) + 2 x (2 x) + 2 x \cdot 3 + 3 (- x^{2}) + 3 (2 x) + 3 \cdot 3 - {(3 - x)}^{2}$

Step 2.1.3.10

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 2.1.3.10.1

Move $x^{2}$ .

$V = x^{4} - 2 x^{3} - 3 x^{2} + 2 \cdot (- 1 (x^{2} x)) + 2 x (2 x) + 2 x \cdot 3 + 3 (- x^{2}) + 3 (2 x) + 3 \cdot 3 - {(3 - x)}^{2}$

Step 2.1.3.10.2

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

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Step 2.1.3.10.2.1

Raise $x$ to the power of $1$ .

$V = x^{4} - 2 x^{3} - 3 x^{2} + 2 \cdot (- 1 (x^{2} x)) + 2 x (2 x) + 2 x \cdot 3 + 3 (- x^{2}) + 3 (2 x) + 3 \cdot 3 - {(3 - x)}^{2}$

Step 2.1.3.10.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$V = x^{4} - 2 x^{3} - 3 x^{2} + 2 \cdot (- 1 x^{2 + 1}) + 2 x (2 x) + 2 x \cdot 3 + 3 (- x^{2}) + 3 (2 x) + 3 \cdot 3 - {(3 - x)}^{2}$

Step 2.1.3.10.3

Add $2$ and $1$ .

$V = x^{4} - 2 x^{3} - 3 x^{2} + 2 \cdot (- 1 x^{3}) + 2 x (2 x) + 2 x \cdot 3 + 3 (- x^{2}) + 3 (2 x) + 3 \cdot 3 - {(3 - x)}^{2}$

Step 2.1.3.11

Multiply $2$ by $- 1$ .

$V = x^{4} - 2 x^{3} - 3 x^{2} - 2 x^{3} + 2 x (2 x) + 2 x \cdot 3 + 3 (- x^{2}) + 3 (2 x) + 3 \cdot 3 - {(3 - x)}^{2}$

Step 2.1.3.12

Rewrite using the commutative property of multiplication.

$V = x^{4} - 2 x^{3} - 3 x^{2} - 2 x^{3} + 2 \cdot (2 x \cdot x) + 2 x \cdot 3 + 3 (- x^{2}) + 3 (2 x) + 3 \cdot 3 - {(3 - x)}^{2}$

Step 2.1.3.13

Multiply $x$ by $x$ by adding the exponents.

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Step 2.1.3.13.1

Move $x$ .

$V = x^{4} - 2 x^{3} - 3 x^{2} - 2 x^{3} + 2 \cdot (2 (x \cdot x)) + 2 x \cdot 3 + 3 (- x^{2}) + 3 (2 x) + 3 \cdot 3 - {(3 - x)}^{2}$

Step 2.1.3.13.2

Multiply $x$ by $x$ .

$V = x^{4} - 2 x^{3} - 3 x^{2} - 2 x^{3} + 2 \cdot (2 x^{2}) + 2 x \cdot 3 + 3 (- x^{2}) + 3 (2 x) + 3 \cdot 3 - {(3 - x)}^{2}$

Step 2.1.3.14

Multiply $2$ by $2$ .

$V = x^{4} - 2 x^{3} - 3 x^{2} - 2 x^{3} + 4 x^{2} + 2 x \cdot 3 + 3 (- x^{2}) + 3 (2 x) + 3 \cdot 3 - {(3 - x)}^{2}$

Step 2.1.3.15

Multiply $3$ by $2$ .

$V = x^{4} - 2 x^{3} - 3 x^{2} - 2 x^{3} + 4 x^{2} + 6 x + 3 (- x^{2}) + 3 (2 x) + 3 \cdot 3 - {(3 - x)}^{2}$

Step 2.1.3.16

Multiply $- 1$ by $3$ .

$V = x^{4} - 2 x^{3} - 3 x^{2} - 2 x^{3} + 4 x^{2} + 6 x - 3 x^{2} + 3 (2 x) + 3 \cdot 3 - {(3 - x)}^{2}$

Step 2.1.3.17

Multiply $2$ by $3$ .

$V = x^{4} - 2 x^{3} - 3 x^{2} - 2 x^{3} + 4 x^{2} + 6 x - 3 x^{2} + 6 x + 3 \cdot 3 - {(3 - x)}^{2}$

Step 2.1.3.18

Multiply $3$ by $3$ .

$V = x^{4} - 2 x^{3} - 3 x^{2} - 2 x^{3} + 4 x^{2} + 6 x - 3 x^{2} + 6 x + 9 - {(3 - x)}^{2}$

Step 2.1.4

Subtract $2 x^{3}$ from $- 2 x^{3}$ .

$V = x^{4} - 4 x^{3} - 3 x^{2} + 4 x^{2} + 6 x - 3 x^{2} + 6 x + 9 - {(3 - x)}^{2}$

Step 2.1.5

Add $- 3 x^{2}$ and $4 x^{2}$ .

$V = x^{4} - 4 x^{3} + x^{2} + 6 x - 3 x^{2} + 6 x + 9 - {(3 - x)}^{2}$

Step 2.1.6

Subtract $3 x^{2}$ from $x^{2}$ .

$V = x^{4} - 4 x^{3} - 2 x^{2} + 6 x + 6 x + 9 - {(3 - x)}^{2}$

Step 2.1.7

Add $6 x$ and $6 x$ .

$V = x^{4} - 4 x^{3} - 2 x^{2} + 12 x + 9 - {(3 - x)}^{2}$

Step 2.1.8

Rewrite ${(3 - x)}^{2}$ as $(3 - x) (3 - x)$ .

$V = x^{4} - 4 x^{3} - 2 x^{2} + 12 x + 9 - ((3 - x) (3 - x))$

Step 2.1.9

Expand $(3 - x) (3 - x)$ using the FOIL Method.

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Step 2.1.9.1

Apply the distributive property.

$V = x^{4} - 4 x^{3} - 2 x^{2} + 12 x + 9 - (3 (3 - x) - x (3 - x))$

Step 2.1.9.2

Apply the distributive property.

$V = x^{4} - 4 x^{3} - 2 x^{2} + 12 x + 9 - (3 \cdot 3 + 3 (- x) - x (3 - x))$

Step 2.1.9.3

Apply the distributive property.

$V = x^{4} - 4 x^{3} - 2 x^{2} + 12 x + 9 - (3 \cdot 3 + 3 (- x) - x \cdot 3 - x (- x))$

Step 2.1.10

Simplify and combine like terms.

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Step 2.1.10.1

Simplify each term.

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Step 2.1.10.1.1

Multiply $3$ by $3$ .

$V = x^{4} - 4 x^{3} - 2 x^{2} + 12 x + 9 - (9 + 3 (- x) - x \cdot 3 - x (- x))$

Step 2.1.10.1.2

Multiply $- 1$ by $3$ .

$V = x^{4} - 4 x^{3} - 2 x^{2} + 12 x + 9 - (9 - 3 x - x \cdot 3 - x (- x))$

Step 2.1.10.1.3

Multiply $3$ by $- 1$ .

$V = x^{4} - 4 x^{3} - 2 x^{2} + 12 x + 9 - (9 - 3 x - 3 x - x (- x))$

Step 2.1.10.1.4

Rewrite using the commutative property of multiplication.

$V = x^{4} - 4 x^{3} - 2 x^{2} + 12 x + 9 - (9 - 3 x - 3 x - 1 \cdot (- 1 x \cdot x))$

Step 2.1.10.1.5

Multiply $x$ by $x$ by adding the exponents.

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Step 2.1.10.1.5.1

Move $x$ .

$V = x^{4} - 4 x^{3} - 2 x^{2} + 12 x + 9 - (9 - 3 x - 3 x - 1 \cdot (- 1 (x \cdot x)))$

Step 2.1.10.1.5.2

Multiply $x$ by $x$ .

$V = x^{4} - 4 x^{3} - 2 x^{2} + 12 x + 9 - (9 - 3 x - 3 x - 1 \cdot (- 1 x^{2}))$

Step 2.1.10.1.6

Multiply $- 1$ by $- 1$ .

$V = x^{4} - 4 x^{3} - 2 x^{2} + 12 x + 9 - (9 - 3 x - 3 x + 1 x^{2})$

Step 2.1.10.1.7

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

$V = x^{4} - 4 x^{3} - 2 x^{2} + 12 x + 9 - (9 - 3 x - 3 x + x^{2})$

Step 2.1.10.2

Subtract $3 x$ from $- 3 x$ .

$V = x^{4} - 4 x^{3} - 2 x^{2} + 12 x + 9 - (9 - 6 x + x^{2})$

Step 2.1.11

Apply the distributive property.

$V = x^{4} - 4 x^{3} - 2 x^{2} + 12 x + 9 - 1 \cdot 9 - (- 6 x) - x^{2}$

Step 2.1.12

Simplify.

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Step 2.1.12.1

Multiply $- 1$ by $9$ .

$V = x^{4} - 4 x^{3} - 2 x^{2} + 12 x + 9 - 9 - (- 6 x) - x^{2}$

Step 2.1.12.2

Multiply $- 6$ by $- 1$ .

$V = x^{4} - 4 x^{3} - 2 x^{2} + 12 x + 9 - 9 + 6 x - x^{2}$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Combine the opposite terms in $x^{4} - 4 x^{3} - 2 x^{2} + 12 x + 9 - 9 + 6 x - x^{2}$ .

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Step 2.2.1.1

Subtract $9$ from $9$ .

$V = x^{4} - 4 x^{3} - 2 x^{2} + 12 x + 0 + 6 x - x^{2}$

Step 2.2.1.2

Add $x^{4} - 4 x^{3} - 2 x^{2} + 12 x$ and $0$ .

$V = x^{4} - 4 x^{3} - 2 x^{2} + 12 x + 6 x - x^{2}$

Step 2.2.2

Subtract $x^{2}$ from $- 2 x^{2}$ .

$V = x^{4} - 4 x^{3} - 3 x^{2} + 12 x + 6 x$

Step 2.2.3

Add $12 x$ and $6 x$ .

$V = x^{4} - 4 x^{3} - 3 x^{2} + 18 x$

Step 3

Split the single integral into multiple integrals.

$V = π (\int_{0}^{3} x^{4} d x + \int_{0}^{3} - 4 x^{3} d x + \int_{0}^{3} - 3 x^{2} d x + \int_{0}^{3} 18 x d x)$

Step 4

By the Power Rule, the integral of $x^{4}$ with respect to $x$ is $\frac{1}{5} x^{5}$ .

$V = π (\frac{1}{5} x^{5}]_{0}^{3} + \int_{0}^{3} - 4 x^{3} d x + \int_{0}^{3} - 3 x^{2} d x + \int_{0}^{3} 18 x d x)$

Step 5

Combine $\frac{1}{5}$ and $x^{5}$ .

$V = π (\frac{x^{5}}{5}]_{0}^{3} + \int_{0}^{3} - 4 x^{3} d x + \int_{0}^{3} - 3 x^{2} d x + \int_{0}^{3} 18 x d x)$

Step 6

Since $- 4$ is constant with respect to $x$ , move $- 4$ out of the integral.

$V = π (\frac{x^{5}}{5}]_{0}^{3} - 4 \int_{0}^{3} x^{3} d x + \int_{0}^{3} - 3 x^{2} d x + \int_{0}^{3} 18 x d x)$

Step 7

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

$V = π (\frac{x^{5}}{5}]_{0}^{3} - 4 (\frac{1}{4} x^{4}]_{0}^{3}) + \int_{0}^{3} - 3 x^{2} d x + \int_{0}^{3} 18 x d x)$

Step 8

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

$V = π (\frac{x^{5}}{5}]_{0}^{3} - 4 (\frac{x^{4}}{4}]_{0}^{3}) + \int_{0}^{3} - 3 x^{2} d x + \int_{0}^{3} 18 x d x)$

Step 9

Since $- 3$ is constant with respect to $x$ , move $- 3$ out of the integral.

$V = π (\frac{x^{5}}{5}]_{0}^{3} - 4 (\frac{x^{4}}{4}]_{0}^{3}) - 3 \int_{0}^{3} x^{2} d x + \int_{0}^{3} 18 x d x)$

Step 10

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

$V = π (\frac{x^{5}}{5}]_{0}^{3} - 4 (\frac{x^{4}}{4}]_{0}^{3}) - 3 (\frac{1}{3} x^{3}]_{0}^{3}) + \int_{0}^{3} 18 x d x)$

Step 11

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

$V = π (\frac{x^{5}}{5}]_{0}^{3} - 4 (\frac{x^{4}}{4}]_{0}^{3}) - 3 (\frac{x^{3}}{3}]_{0}^{3}) + \int_{0}^{3} 18 x d x)$

Step 12

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

$V = π (\frac{x^{5}}{5}]_{0}^{3} - 4 (\frac{x^{4}}{4}]_{0}^{3}) - 3 (\frac{x^{3}}{3}]_{0}^{3}) + 18 \int_{0}^{3} x d x)$

Step 13

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

$V = π (\frac{x^{5}}{5}]_{0}^{3} - 4 (\frac{x^{4}}{4}]_{0}^{3}) - 3 (\frac{x^{3}}{3}]_{0}^{3}) + 18 (\frac{1}{2} x^{2}]_{0}^{3}))$

Step 14

Simplify the answer.

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Step 14.1

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

$V = π (\frac{x^{5}}{5}]_{0}^{3} - 4 (\frac{x^{4}}{4}]_{0}^{3}) - 3 (\frac{x^{3}}{3}]_{0}^{3}) + 18 (\frac{x^{2}}{2}]_{0}^{3}))$

Step 14.2

Substitute and simplify.

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Step 14.2.1

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

$V = π ((\frac{3^{5}}{5}) - \frac{0^{5}}{5} - 4 (\frac{x^{4}}{4}]_{0}^{3}) - 3 (\frac{x^{3}}{3}]_{0}^{3}) + 18 (\frac{x^{2}}{2}]_{0}^{3}))$

Step 14.2.2

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

$V = π (\frac{3^{5}}{5} - \frac{0^{5}}{5} - 4 (\frac{3^{4}}{4} - \frac{0^{4}}{4}) - 3 (\frac{x^{3}}{3}]_{0}^{3}) + 18 (\frac{x^{2}}{2}]_{0}^{3}))$

Step 14.2.3

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

$V = π (\frac{3^{5}}{5} - \frac{0^{5}}{5} - 4 (\frac{3^{4}}{4} - \frac{0^{4}}{4}) - 3 ((\frac{3^{3}}{3}) - \frac{0^{3}}{3}) + 18 (\frac{x^{2}}{2}]_{0}^{3}))$

Step 14.2.4

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

$V = π (\frac{3^{5}}{5} - \frac{0^{5}}{5} - 4 (\frac{3^{4}}{4} - \frac{0^{4}}{4}) - 3 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) + 18 (\frac{3^{2}}{2} - \frac{0^{2}}{2}))$

Step 14.2.5

Simplify.

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Step 14.2.5.1

Raise $3$ to the power of $5$ .

$V = π (\frac{243}{5} - \frac{0^{5}}{5} - 4 (\frac{3^{4}}{4} - \frac{0^{4}}{4}) - 3 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) + 18 (\frac{3^{2}}{2} - \frac{0^{2}}{2}))$

Step 14.2.5.2

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

$V = π (\frac{243}{5} - \frac{0}{5} - 4 (\frac{3^{4}}{4} - \frac{0^{4}}{4}) - 3 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) + 18 (\frac{3^{2}}{2} - \frac{0^{2}}{2}))$

Step 14.2.5.3

Cancel the common factor of $0$ and $5$ .

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Step 14.2.5.3.1

Factor $5$ out of $0$ .

$V = π (\frac{243}{5} - \frac{5 (0)}{5} - 4 (\frac{3^{4}}{4} - \frac{0^{4}}{4}) - 3 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) + 18 (\frac{3^{2}}{2} - \frac{0^{2}}{2}))$

Step 14.2.5.3.2

Cancel the common factors.

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Step 14.2.5.3.2.1

Factor $5$ out of $5$ .

$V = π (\frac{243}{5} - \frac{5 \cdot 0}{5 \cdot 1} - 4 (\frac{3^{4}}{4} - \frac{0^{4}}{4}) - 3 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) + 18 (\frac{3^{2}}{2} - \frac{0^{2}}{2}))$

Step 14.2.5.3.2.2

Cancel the common factor.

$V = π (\frac{243}{5} - \frac{5 \cdot 0}{5 \cdot 1} - 4 (\frac{3^{4}}{4} - \frac{0^{4}}{4}) - 3 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) + 18 (\frac{3^{2}}{2} - \frac{0^{2}}{2}))$

Step 14.2.5.3.2.3

Rewrite the expression.

$V = π (\frac{243}{5} - \frac{0}{1} - 4 (\frac{3^{4}}{4} - \frac{0^{4}}{4}) - 3 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) + 18 (\frac{3^{2}}{2} - \frac{0^{2}}{2}))$

Step 14.2.5.3.2.4

Divide $0$ by $1$ .

$V = π (\frac{243}{5} - 0 - 4 (\frac{3^{4}}{4} - \frac{0^{4}}{4}) - 3 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) + 18 (\frac{3^{2}}{2} - \frac{0^{2}}{2}))$

Step 14.2.5.4

Multiply $- 1$ by $0$ .

$V = π (\frac{243}{5} + 0 - 4 (\frac{3^{4}}{4} - \frac{0^{4}}{4}) - 3 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) + 18 (\frac{3^{2}}{2} - \frac{0^{2}}{2}))$

Step 14.2.5.5

Add $\frac{243}{5}$ and $0$ .

$V = π (\frac{243}{5} - 4 (\frac{3^{4}}{4} - \frac{0^{4}}{4}) - 3 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) + 18 (\frac{3^{2}}{2} - \frac{0^{2}}{2}))$

Step 14.2.5.6

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

$V = π (\frac{243}{5} - 4 (\frac{81}{4} - \frac{0^{4}}{4}) - 3 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) + 18 (\frac{3^{2}}{2} - \frac{0^{2}}{2}))$

Step 14.2.5.7

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

$V = π (\frac{243}{5} - 4 (\frac{81}{4} - \frac{0}{4}) - 3 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) + 18 (\frac{3^{2}}{2} - \frac{0^{2}}{2}))$

Step 14.2.5.8

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

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Step 14.2.5.8.1

Factor $4$ out of $0$ .

$V = π (\frac{243}{5} - 4 (\frac{81}{4} - \frac{4 (0)}{4}) - 3 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) + 18 (\frac{3^{2}}{2} - \frac{0^{2}}{2}))$

Step 14.2.5.8.2

Cancel the common factors.

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Step 14.2.5.8.2.1

Factor $4$ out of $4$ .

$V = π (\frac{243}{5} - 4 (\frac{81}{4} - \frac{4 \cdot 0}{4 \cdot 1}) - 3 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) + 18 (\frac{3^{2}}{2} - \frac{0^{2}}{2}))$

Step 14.2.5.8.2.2

Cancel the common factor.

$V = π (\frac{243}{5} - 4 (\frac{81}{4} - \frac{4 \cdot 0}{4 \cdot 1}) - 3 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) + 18 (\frac{3^{2}}{2} - \frac{0^{2}}{2}))$

Step 14.2.5.8.2.3

Rewrite the expression.

$V = π (\frac{243}{5} - 4 (\frac{81}{4} - \frac{0}{1}) - 3 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) + 18 (\frac{3^{2}}{2} - \frac{0^{2}}{2}))$

Step 14.2.5.8.2.4

Divide $0$ by $1$ .

$V = π (\frac{243}{5} - 4 (\frac{81}{4} - 0) - 3 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) + 18 (\frac{3^{2}}{2} - \frac{0^{2}}{2}))$

Step 14.2.5.9

Multiply $- 1$ by $0$ .

$V = π (\frac{243}{5} - 4 (\frac{81}{4} + 0) - 3 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) + 18 (\frac{3^{2}}{2} - \frac{0^{2}}{2}))$

Step 14.2.5.10

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

$V = π (\frac{243}{5} - 4 (\frac{81}{4}) - 3 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) + 18 (\frac{3^{2}}{2} - \frac{0^{2}}{2}))$

Step 14.2.5.11

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

$V = π (\frac{243}{5} + \frac{- 4 \cdot 81}{4} - 3 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) + 18 (\frac{3^{2}}{2} - \frac{0^{2}}{2}))$

Step 14.2.5.12

Multiply $- 4$ by $81$ .

$V = π (\frac{243}{5} + \frac{- 324}{4} - 3 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) + 18 (\frac{3^{2}}{2} - \frac{0^{2}}{2}))$

Step 14.2.5.13

Cancel the common factor of $- 324$ and $4$ .

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Step 14.2.5.13.1

Factor $4$ out of $- 324$ .

$V = π (\frac{243}{5} + \frac{4 \cdot - 81}{4} - 3 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) + 18 (\frac{3^{2}}{2} - \frac{0^{2}}{2}))$

Step 14.2.5.13.2

Cancel the common factors.

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Step 14.2.5.13.2.1

Factor $4$ out of $4$ .

$V = π (\frac{243}{5} + \frac{4 \cdot - 81}{4 (1)} - 3 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) + 18 (\frac{3^{2}}{2} - \frac{0^{2}}{2}))$

Step 14.2.5.13.2.2

Cancel the common factor.

$V = π (\frac{243}{5} + \frac{4 \cdot - 81}{4 \cdot 1} - 3 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) + 18 (\frac{3^{2}}{2} - \frac{0^{2}}{2}))$

Step 14.2.5.13.2.3

Rewrite the expression.

$V = π (\frac{243}{5} + \frac{- 81}{1} - 3 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) + 18 (\frac{3^{2}}{2} - \frac{0^{2}}{2}))$

Step 14.2.5.13.2.4

Divide $- 81$ by $1$ .

$V = π (\frac{243}{5} - 81 - 3 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) + 18 (\frac{3^{2}}{2} - \frac{0^{2}}{2}))$

Step 14.2.5.14

To write $- 81$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$V = π (\frac{243}{5} - 81 \cdot \frac{5}{5} - 3 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) + 18 (\frac{3^{2}}{2} - \frac{0^{2}}{2}))$

Step 14.2.5.15

Combine $- 81$ and $\frac{5}{5}$ .

$V = π (\frac{243}{5} + \frac{- 81 \cdot 5}{5} - 3 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) + 18 (\frac{3^{2}}{2} - \frac{0^{2}}{2}))$

Step 14.2.5.16

Combine the numerators over the common denominator.

$V = π (\frac{243 - 81 \cdot 5}{5} - 3 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) + 18 (\frac{3^{2}}{2} - \frac{0^{2}}{2}))$

Step 14.2.5.17

Simplify the numerator.

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Step 14.2.5.17.1

Multiply $- 81$ by $5$ .

$V = π (\frac{243 - 405}{5} - 3 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) + 18 (\frac{3^{2}}{2} - \frac{0^{2}}{2}))$

Step 14.2.5.17.2

Subtract $405$ from $243$ .

$V = π (\frac{- 162}{5} - 3 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) + 18 (\frac{3^{2}}{2} - \frac{0^{2}}{2}))$

Step 14.2.5.18

Move the negative in front of the fraction.

$V = π (- \frac{162}{5} - 3 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) + 18 (\frac{3^{2}}{2} - \frac{0^{2}}{2}))$

Step 14.2.5.19

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

$V = π (- \frac{162}{5} - 3 (\frac{27}{3} - \frac{0^{3}}{3}) + 18 (\frac{3^{2}}{2} - \frac{0^{2}}{2}))$

Step 14.2.5.20

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

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Step 14.2.5.20.1

Factor $3$ out of $27$ .

$V = π (- \frac{162}{5} - 3 (\frac{3 \cdot 9}{3} - \frac{0^{3}}{3}) + 18 (\frac{3^{2}}{2} - \frac{0^{2}}{2}))$

Step 14.2.5.20.2

Cancel the common factors.

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Step 14.2.5.20.2.1

Factor $3$ out of $3$ .

$V = π (- \frac{162}{5} - 3 (\frac{3 \cdot 9}{3 (1)} - \frac{0^{3}}{3}) + 18 (\frac{3^{2}}{2} - \frac{0^{2}}{2}))$

Step 14.2.5.20.2.2

Cancel the common factor.

$V = π (- \frac{162}{5} - 3 (\frac{3 \cdot 9}{3 \cdot 1} - \frac{0^{3}}{3}) + 18 (\frac{3^{2}}{2} - \frac{0^{2}}{2}))$

Step 14.2.5.20.2.3

Rewrite the expression.

$V = π (- \frac{162}{5} - 3 (\frac{9}{1} - \frac{0^{3}}{3}) + 18 (\frac{3^{2}}{2} - \frac{0^{2}}{2}))$

Step 14.2.5.20.2.4

Divide $9$ by $1$ .

$V = π (- \frac{162}{5} - 3 (9 - \frac{0^{3}}{3}) + 18 (\frac{3^{2}}{2} - \frac{0^{2}}{2}))$

Step 14.2.5.21

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

$V = π (- \frac{162}{5} - 3 (9 - \frac{0}{3}) + 18 (\frac{3^{2}}{2} - \frac{0^{2}}{2}))$

Step 14.2.5.22

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

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Step 14.2.5.22.1

Factor $3$ out of $0$ .

$V = π (- \frac{162}{5} - 3 (9 - \frac{3 (0)}{3}) + 18 (\frac{3^{2}}{2} - \frac{0^{2}}{2}))$

Step 14.2.5.22.2

Cancel the common factors.

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Step 14.2.5.22.2.1

Factor $3$ out of $3$ .

$V = π (- \frac{162}{5} - 3 (9 - \frac{3 \cdot 0}{3 \cdot 1}) + 18 (\frac{3^{2}}{2} - \frac{0^{2}}{2}))$

Step 14.2.5.22.2.2

Cancel the common factor.

$V = π (- \frac{162}{5} - 3 (9 - \frac{3 \cdot 0}{3 \cdot 1}) + 18 (\frac{3^{2}}{2} - \frac{0^{2}}{2}))$

Step 14.2.5.22.2.3

Rewrite the expression.

$V = π (- \frac{162}{5} - 3 (9 - \frac{0}{1}) + 18 (\frac{3^{2}}{2} - \frac{0^{2}}{2}))$

Step 14.2.5.22.2.4

Divide $0$ by $1$ .

$V = π (- \frac{162}{5} - 3 (9 - 0) + 18 (\frac{3^{2}}{2} - \frac{0^{2}}{2}))$

Step 14.2.5.23

Multiply $- 1$ by $0$ .

$V = π (- \frac{162}{5} - 3 (9 + 0) + 18 (\frac{3^{2}}{2} - \frac{0^{2}}{2}))$

Step 14.2.5.24

Add $9$ and $0$ .

$V = π (- \frac{162}{5} - 3 \cdot 9 + 18 (\frac{3^{2}}{2} - \frac{0^{2}}{2}))$

Step 14.2.5.25

Multiply $- 3$ by $9$ .

$V = π (- \frac{162}{5} - 27 + 18 (\frac{3^{2}}{2} - \frac{0^{2}}{2}))$

Step 14.2.5.26

To write $- 27$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$V = π (- \frac{162}{5} - 27 \cdot \frac{5}{5} + 18 (\frac{3^{2}}{2} - \frac{0^{2}}{2}))$

Step 14.2.5.27

Combine $- 27$ and $\frac{5}{5}$ .

$V = π (- \frac{162}{5} + \frac{- 27 \cdot 5}{5} + 18 (\frac{3^{2}}{2} - \frac{0^{2}}{2}))$

Step 14.2.5.28

Combine the numerators over the common denominator.

$V = π (\frac{- 162 - 27 \cdot 5}{5} + 18 (\frac{3^{2}}{2} - \frac{0^{2}}{2}))$

Step 14.2.5.29

Simplify the numerator.

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Step 14.2.5.29.1

Multiply $- 27$ by $5$ .

$V = π (\frac{- 162 - 135}{5} + 18 (\frac{3^{2}}{2} - \frac{0^{2}}{2}))$

Step 14.2.5.29.2

Subtract $135$ from $- 162$ .

$V = π (\frac{- 297}{5} + 18 (\frac{3^{2}}{2} - \frac{0^{2}}{2}))$

Step 14.2.5.30

Move the negative in front of the fraction.

$V = π (- \frac{297}{5} + 18 (\frac{3^{2}}{2} - \frac{0^{2}}{2}))$

Step 14.2.5.31

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

$V = π (- \frac{297}{5} + 18 (\frac{9}{2} - \frac{0^{2}}{2}))$

Step 14.2.5.32

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

$V = π (- \frac{297}{5} + 18 (\frac{9}{2} - \frac{0}{2}))$

Step 14.2.5.33

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

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Step 14.2.5.33.1

Factor $2$ out of $0$ .

$V = π (- \frac{297}{5} + 18 (\frac{9}{2} - \frac{2 (0)}{2}))$

Step 14.2.5.33.2

Cancel the common factors.

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Step 14.2.5.33.2.1

Factor $2$ out of $2$ .

$V = π (- \frac{297}{5} + 18 (\frac{9}{2} - \frac{2 \cdot 0}{2 \cdot 1}))$

Step 14.2.5.33.2.2

Cancel the common factor.

$V = π (- \frac{297}{5} + 18 (\frac{9}{2} - \frac{2 \cdot 0}{2 \cdot 1}))$

Step 14.2.5.33.2.3

Rewrite the expression.

$V = π (- \frac{297}{5} + 18 (\frac{9}{2} - \frac{0}{1}))$

Step 14.2.5.33.2.4

Divide $0$ by $1$ .

$V = π (- \frac{297}{5} + 18 (\frac{9}{2} - 0))$

Step 14.2.5.34

Multiply $- 1$ by $0$ .

$V = π (- \frac{297}{5} + 18 (\frac{9}{2} + 0))$

Step 14.2.5.35

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

$V = π (- \frac{297}{5} + 18 (\frac{9}{2}))$

Step 14.2.5.36

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

$V = π (- \frac{297}{5} + \frac{18 \cdot 9}{2})$

Step 14.2.5.37

Multiply $18$ by $9$ .

$V = π (- \frac{297}{5} + \frac{162}{2})$

Step 14.2.5.38

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

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Step 14.2.5.38.1

Factor $2$ out of $162$ .

$V = π (- \frac{297}{5} + \frac{2 \cdot 81}{2})$

Step 14.2.5.38.2

Cancel the common factors.

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Step 14.2.5.38.2.1

Factor $2$ out of $2$ .

$V = π (- \frac{297}{5} + \frac{2 \cdot 81}{2 (1)})$

Step 14.2.5.38.2.2

Cancel the common factor.

$V = π (- \frac{297}{5} + \frac{2 \cdot 81}{2 \cdot 1})$

Step 14.2.5.38.2.3

Rewrite the expression.

$V = π (- \frac{297}{5} + \frac{81}{1})$

Step 14.2.5.38.2.4

Divide $81$ by $1$ .

$V = π (- \frac{297}{5} + 81)$

Step 14.2.5.39

To write $81$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$V = π (- \frac{297}{5} + 81 \cdot \frac{5}{5})$

Step 14.2.5.40

Combine $81$ and $\frac{5}{5}$ .

$V = π (- \frac{297}{5} + \frac{81 \cdot 5}{5})$

Step 14.2.5.41

Combine the numerators over the common denominator.

$V = π (\frac{- 297 + 81 \cdot 5}{5})$

Step 14.2.5.42

Simplify the numerator.

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Step 14.2.5.42.1

Multiply $81$ by $5$ .

$V = π (\frac{- 297 + 405}{5})$

Step 14.2.5.42.2

Add $- 297$ and $405$ .

$V = π (\frac{108}{5})$

Step 14.2.5.43

Combine $π$ and $\frac{108}{5}$ .

$V = \frac{π \cdot 108}{5}$

Step 14.2.5.44

Move $108$ to the left of $π$ .

$V = \frac{108 π}{5}$

Step 15

The result can be shown in multiple forms.

Exact Form:

$V = \frac{108 π}{5}$

Decimal Form:

$V = 67.85840131 \dots$

Step 16