Calculus Examples

$y = - x^{2} + 14 x - 45$ , $y = 0$

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 $A = π r^{2}$ .

$V = π \int_{5}^{9} {(f (x))}^{2} d x$ where $f (x) = - x^{2} + 14 x - 45$

Step 2

Simplify the integrand.

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Step 2.1

Rewrite ${(- x^{2} + 14 x - 45)}^{2}$ as $(- x^{2} + 14 x - 45) (- x^{2} + 14 x - 45)$ .

$V = (- x^{2} + 14 x - 45) (- x^{2} + 14 x - 45)$

Step 2.2

Expand $(- x^{2} + 14 x - 45) (- x^{2} + 14 x - 45)$ by multiplying each term in the first expression by each term in the second expression.

$V = - x^{2} (- x^{2}) - x^{2} (14 x) - x^{2} \cdot - 45 + 14 x (- x^{2}) + 14 x (14 x) + 14 x \cdot - 45 - 45 (- x^{2}) - 45 (14 x) - 45 \cdot - 45$

Step 2.3

Simplify terms.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Rewrite using the commutative property of multiplication.

$V = - 1 \cdot (- 1 x^{2} x^{2}) - x^{2} (14 x) - x^{2} \cdot - 45 + 14 x (- x^{2}) + 14 x (14 x) + 14 x \cdot - 45 - 45 (- x^{2}) - 45 (14 x) - 45 \cdot - 45$

Step 2.3.1.2

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

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Step 2.3.1.2.1

Move $x^{2}$ .

$V = - 1 \cdot (- 1 (x^{2} x^{2})) - x^{2} (14 x) - x^{2} \cdot - 45 + 14 x (- x^{2}) + 14 x (14 x) + 14 x \cdot - 45 - 45 (- x^{2}) - 45 (14 x) - 45 \cdot - 45$

Step 2.3.1.2.2

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

$V = - 1 \cdot (- 1 x^{2 + 2}) - x^{2} (14 x) - x^{2} \cdot - 45 + 14 x (- x^{2}) + 14 x (14 x) + 14 x \cdot - 45 - 45 (- x^{2}) - 45 (14 x) - 45 \cdot - 45$

Step 2.3.1.2.3

Add $2$ and $2$ .

$V = - 1 \cdot (- 1 x^{4}) - x^{2} (14 x) - x^{2} \cdot - 45 + 14 x (- x^{2}) + 14 x (14 x) + 14 x \cdot - 45 - 45 (- x^{2}) - 45 (14 x) - 45 \cdot - 45$

Step 2.3.1.3

Multiply $- 1$ by $- 1$ .

$V = 1 x^{4} - x^{2} (14 x) - x^{2} \cdot - 45 + 14 x (- x^{2}) + 14 x (14 x) + 14 x \cdot - 45 - 45 (- x^{2}) - 45 (14 x) - 45 \cdot - 45$

Step 2.3.1.4

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

$V = x^{4} - x^{2} (14 x) - x^{2} \cdot - 45 + 14 x (- x^{2}) + 14 x (14 x) + 14 x \cdot - 45 - 45 (- x^{2}) - 45 (14 x) - 45 \cdot - 45$

Step 2.3.1.5

Rewrite using the commutative property of multiplication.

$V = x^{4} - 1 \cdot (14 x^{2} x) - x^{2} \cdot - 45 + 14 x (- x^{2}) + 14 x (14 x) + 14 x \cdot - 45 - 45 (- x^{2}) - 45 (14 x) - 45 \cdot - 45$

Step 2.3.1.6

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

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Step 2.3.1.6.1

Move $x$ .

$V = x^{4} - 1 \cdot (14 (x \cdot x^{2})) - x^{2} \cdot - 45 + 14 x (- x^{2}) + 14 x (14 x) + 14 x \cdot - 45 - 45 (- x^{2}) - 45 (14 x) - 45 \cdot - 45$

Step 2.3.1.6.2

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

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Step 2.3.1.6.2.1

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

$V = x^{4} - 1 \cdot (14 (x \cdot x^{2})) - x^{2} \cdot - 45 + 14 x (- x^{2}) + 14 x (14 x) + 14 x \cdot - 45 - 45 (- x^{2}) - 45 (14 x) - 45 \cdot - 45$

Step 2.3.1.6.2.2

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

$V = x^{4} - 1 \cdot (14 x^{1 + 2}) - x^{2} \cdot - 45 + 14 x (- x^{2}) + 14 x (14 x) + 14 x \cdot - 45 - 45 (- x^{2}) - 45 (14 x) - 45 \cdot - 45$

Step 2.3.1.6.3

Add $1$ and $2$ .

$V = x^{4} - 1 \cdot (14 x^{3}) - x^{2} \cdot - 45 + 14 x (- x^{2}) + 14 x (14 x) + 14 x \cdot - 45 - 45 (- x^{2}) - 45 (14 x) - 45 \cdot - 45$

Step 2.3.1.7

Multiply $- 1$ by $14$ .

$V = x^{4} - 14 x^{3} - x^{2} \cdot - 45 + 14 x (- x^{2}) + 14 x (14 x) + 14 x \cdot - 45 - 45 (- x^{2}) - 45 (14 x) - 45 \cdot - 45$

Step 2.3.1.8

Multiply $- 45$ by $- 1$ .

$V = x^{4} - 14 x^{3} + 45 x^{2} + 14 x (- x^{2}) + 14 x (14 x) + 14 x \cdot - 45 - 45 (- x^{2}) - 45 (14 x) - 45 \cdot - 45$

Step 2.3.1.9

Rewrite using the commutative property of multiplication.

$V = x^{4} - 14 x^{3} + 45 x^{2} + 14 \cdot (- 1 x \cdot x^{2}) + 14 x (14 x) + 14 x \cdot - 45 - 45 (- x^{2}) - 45 (14 x) - 45 \cdot - 45$

Step 2.3.1.10

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

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Step 2.3.1.10.1

Move $x^{2}$ .

$V = x^{4} - 14 x^{3} + 45 x^{2} + 14 \cdot (- 1 (x^{2} x)) + 14 x (14 x) + 14 x \cdot - 45 - 45 (- x^{2}) - 45 (14 x) - 45 \cdot - 45$

Step 2.3.1.10.2

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

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Step 2.3.1.10.2.1

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

$V = x^{4} - 14 x^{3} + 45 x^{2} + 14 \cdot (- 1 (x^{2} x)) + 14 x (14 x) + 14 x \cdot - 45 - 45 (- x^{2}) - 45 (14 x) - 45 \cdot - 45$

Step 2.3.1.10.2.2

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

$V = x^{4} - 14 x^{3} + 45 x^{2} + 14 \cdot (- 1 x^{2 + 1}) + 14 x (14 x) + 14 x \cdot - 45 - 45 (- x^{2}) - 45 (14 x) - 45 \cdot - 45$

Step 2.3.1.10.3

Add $2$ and $1$ .

$V = x^{4} - 14 x^{3} + 45 x^{2} + 14 \cdot (- 1 x^{3}) + 14 x (14 x) + 14 x \cdot - 45 - 45 (- x^{2}) - 45 (14 x) - 45 \cdot - 45$

Step 2.3.1.11

Multiply $14$ by $- 1$ .

$V = x^{4} - 14 x^{3} + 45 x^{2} - 14 x^{3} + 14 x (14 x) + 14 x \cdot - 45 - 45 (- x^{2}) - 45 (14 x) - 45 \cdot - 45$

Step 2.3.1.12

Rewrite using the commutative property of multiplication.

$V = x^{4} - 14 x^{3} + 45 x^{2} - 14 x^{3} + 14 \cdot (14 x \cdot x) + 14 x \cdot - 45 - 45 (- x^{2}) - 45 (14 x) - 45 \cdot - 45$

Step 2.3.1.13

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

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Step 2.3.1.13.1

Move $x$ .

$V = x^{4} - 14 x^{3} + 45 x^{2} - 14 x^{3} + 14 \cdot (14 (x \cdot x)) + 14 x \cdot - 45 - 45 (- x^{2}) - 45 (14 x) - 45 \cdot - 45$

Step 2.3.1.13.2

Multiply $x$ by $x$ .

$V = x^{4} - 14 x^{3} + 45 x^{2} - 14 x^{3} + 14 \cdot (14 x^{2}) + 14 x \cdot - 45 - 45 (- x^{2}) - 45 (14 x) - 45 \cdot - 45$

Step 2.3.1.14

Multiply $14$ by $14$ .

$V = x^{4} - 14 x^{3} + 45 x^{2} - 14 x^{3} + 196 x^{2} + 14 x \cdot - 45 - 45 (- x^{2}) - 45 (14 x) - 45 \cdot - 45$

Step 2.3.1.15

Multiply $- 45$ by $14$ .

$V = x^{4} - 14 x^{3} + 45 x^{2} - 14 x^{3} + 196 x^{2} - 630 x - 45 (- x^{2}) - 45 (14 x) - 45 \cdot - 45$

Step 2.3.1.16

Multiply $- 1$ by $- 45$ .

$V = x^{4} - 14 x^{3} + 45 x^{2} - 14 x^{3} + 196 x^{2} - 630 x + 45 x^{2} - 45 (14 x) - 45 \cdot - 45$

Step 2.3.1.17

Multiply $14$ by $- 45$ .

$V = x^{4} - 14 x^{3} + 45 x^{2} - 14 x^{3} + 196 x^{2} - 630 x + 45 x^{2} - 630 x - 45 \cdot - 45$

Step 2.3.1.18

Multiply $- 45$ by $- 45$ .

$V = x^{4} - 14 x^{3} + 45 x^{2} - 14 x^{3} + 196 x^{2} - 630 x + 45 x^{2} - 630 x + 2025$

Step 2.3.2

Simplify by adding terms.

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Step 2.3.2.1

Subtract $14 x^{3}$ from $- 14 x^{3}$ .

$V = x^{4} - 28 x^{3} + 45 x^{2} + 196 x^{2} - 630 x + 45 x^{2} - 630 x + 2025$

Step 2.3.2.2

Add $45 x^{2}$ and $196 x^{2}$ .

$V = x^{4} - 28 x^{3} + 241 x^{2} - 630 x + 45 x^{2} - 630 x + 2025$

Step 2.3.2.3

Add $241 x^{2}$ and $45 x^{2}$ .

$V = x^{4} - 28 x^{3} + 286 x^{2} - 630 x - 630 x + 2025$

Step 2.3.2.4

Subtract $630 x$ from $- 630 x$ .

$V = x^{4} - 28 x^{3} + 286 x^{2} - 1260 x + 2025$

Step 3

Split the single integral into multiple integrals.

$V = π (\int_{5}^{9} x^{4} d x + \int_{5}^{9} - 28 x^{3} d x + \int_{5}^{9} 286 x^{2} d x + \int_{5}^{9} - 1260 x d x + \int_{5}^{9} 2025 d x)$

Step 4

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

$V = π (\frac{1}{5} x^{5}]_{5}^{9} + \int_{5}^{9} - 28 x^{3} d x + \int_{5}^{9} 286 x^{2} d x + \int_{5}^{9} - 1260 x d x + \int_{5}^{9} 2025 d x)$

Step 5

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

$V = π (\frac{x^{5}}{5}]_{5}^{9} + \int_{5}^{9} - 28 x^{3} d x + \int_{5}^{9} 286 x^{2} d x + \int_{5}^{9} - 1260 x d x + \int_{5}^{9} 2025 d x)$

Step 6

Since $- 28$ is constant with respect to $x$ , move $- 28$ out of the integral.

$V = π (\frac{x^{5}}{5}]_{5}^{9} - 28 \int_{5}^{9} x^{3} d x + \int_{5}^{9} 286 x^{2} d x + \int_{5}^{9} - 1260 x d x + \int_{5}^{9} 2025 d x)$

Step 7

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

$V = π (\frac{x^{5}}{5}]_{5}^{9} - 28 (\frac{1}{4} x^{4}]_{5}^{9}) + \int_{5}^{9} 286 x^{2} d x + \int_{5}^{9} - 1260 x d x + \int_{5}^{9} 2025 d x)$

Step 8

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

$V = π (\frac{x^{5}}{5}]_{5}^{9} - 28 (\frac{x^{4}}{4}]_{5}^{9}) + \int_{5}^{9} 286 x^{2} d x + \int_{5}^{9} - 1260 x d x + \int_{5}^{9} 2025 d x)$

Step 9

Since $286$ is constant with respect to $x$ , move $286$ out of the integral.

$V = π (\frac{x^{5}}{5}]_{5}^{9} - 28 (\frac{x^{4}}{4}]_{5}^{9}) + 286 \int_{5}^{9} x^{2} d x + \int_{5}^{9} - 1260 x d x + \int_{5}^{9} 2025 d x)$

Step 10

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

$V = π (\frac{x^{5}}{5}]_{5}^{9} - 28 (\frac{x^{4}}{4}]_{5}^{9}) + 286 (\frac{1}{3} x^{3}]_{5}^{9}) + \int_{5}^{9} - 1260 x d x + \int_{5}^{9} 2025 d x)$

Step 11

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

$V = π (\frac{x^{5}}{5}]_{5}^{9} - 28 (\frac{x^{4}}{4}]_{5}^{9}) + 286 (\frac{x^{3}}{3}]_{5}^{9}) + \int_{5}^{9} - 1260 x d x + \int_{5}^{9} 2025 d x)$

Step 12

Since $- 1260$ is constant with respect to $x$ , move $- 1260$ out of the integral.

$V = π (\frac{x^{5}}{5}]_{5}^{9} - 28 (\frac{x^{4}}{4}]_{5}^{9}) + 286 (\frac{x^{3}}{3}]_{5}^{9}) - 1260 \int_{5}^{9} x d x + \int_{5}^{9} 2025 d x)$

Step 13

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

$V = π (\frac{x^{5}}{5}]_{5}^{9} - 28 (\frac{x^{4}}{4}]_{5}^{9}) + 286 (\frac{x^{3}}{3}]_{5}^{9}) - 1260 (\frac{1}{2} x^{2}]_{5}^{9}) + \int_{5}^{9} 2025 d x)$

Step 14

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

$V = π (\frac{x^{5}}{5}]_{5}^{9} - 28 (\frac{x^{4}}{4}]_{5}^{9}) + 286 (\frac{x^{3}}{3}]_{5}^{9}) - 1260 (\frac{x^{2}}{2}]_{5}^{9}) + \int_{5}^{9} 2025 d x)$

Step 15

Apply the constant rule.

$V = π (\frac{x^{5}}{5}]_{5}^{9} - 28 (\frac{x^{4}}{4}]_{5}^{9}) + 286 (\frac{x^{3}}{3}]_{5}^{9}) - 1260 (\frac{x^{2}}{2}]_{5}^{9}) + 2025 x]_{5}^{9})$

Step 16

Simplify the answer.

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Step 16.1

Combine $\frac{x^{5}}{5}]_{5}^{9}$ and $2025 x]_{5}^{9}$ .

$V = π (\frac{x^{5}}{5} + 2025 x]_{5}^{9} - 28 (\frac{x^{4}}{4}]_{5}^{9}) + 286 (\frac{x^{3}}{3}]_{5}^{9}) - 1260 (\frac{x^{2}}{2}]_{5}^{9}))$

Step 16.2

Substitute and simplify.

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Step 16.2.1

Evaluate $\frac{x^{5}}{5} + 2025 x$ at $9$ and at $5$ .

$V = π ((\frac{9^{5}}{5} + 2025 \cdot 9) - (\frac{5^{5}}{5} + 2025 \cdot 5) - 28 (\frac{x^{4}}{4}]_{5}^{9}) + 286 (\frac{x^{3}}{3}]_{5}^{9}) - 1260 (\frac{x^{2}}{2}]_{5}^{9}))$

Step 16.2.2

Evaluate $\frac{x^{4}}{4}$ at $9$ and at $5$ .

$V = π ((\frac{9^{5}}{5} + 2025 \cdot 9) - (\frac{5^{5}}{5} + 2025 \cdot 5) - 28 ((\frac{9^{4}}{4}) - \frac{5^{4}}{4}) + 286 (\frac{x^{3}}{3}]_{5}^{9}) - 1260 (\frac{x^{2}}{2}]_{5}^{9}))$

Step 16.2.3

Evaluate $\frac{x^{3}}{3}$ at $9$ and at $5$ .

$V = π ((\frac{9^{5}}{5} + 2025 \cdot 9) - (\frac{5^{5}}{5} + 2025 \cdot 5) - 28 (\frac{9^{4}}{4} - \frac{5^{4}}{4}) + 286 (\frac{9^{3}}{3} - \frac{5^{3}}{3}) - 1260 (\frac{x^{2}}{2}]_{5}^{9}))$

Step 16.2.4

Evaluate $\frac{x^{2}}{2}$ at $9$ and at $5$ .

$V = π ((\frac{9^{5}}{5} + 2025 \cdot 9) - (\frac{5^{5}}{5} + 2025 \cdot 5) - 28 (\frac{9^{4}}{4} - \frac{5^{4}}{4}) + 286 (\frac{9^{3}}{3} - \frac{5^{3}}{3}) - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5

Simplify.

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Step 16.2.5.1

Raise $9$ to the power of $5$ .

$V = π (\frac{59049}{5} + 2025 \cdot 9 - (\frac{5^{5}}{5} + 2025 \cdot 5) - 28 (\frac{9^{4}}{4} - \frac{5^{4}}{4}) + 286 (\frac{9^{3}}{3} - \frac{5^{3}}{3}) - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.2

Multiply $2025$ by $9$ .

$V = π (\frac{59049}{5} + 18225 - (\frac{5^{5}}{5} + 2025 \cdot 5) - 28 (\frac{9^{4}}{4} - \frac{5^{4}}{4}) + 286 (\frac{9^{3}}{3} - \frac{5^{3}}{3}) - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.3

To write $18225$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$V = π (\frac{59049}{5} + 18225 \cdot \frac{5}{5} - (\frac{5^{5}}{5} + 2025 \cdot 5) - 28 (\frac{9^{4}}{4} - \frac{5^{4}}{4}) + 286 (\frac{9^{3}}{3} - \frac{5^{3}}{3}) - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.4

Combine $18225$ and $\frac{5}{5}$ .

$V = π (\frac{59049}{5} + \frac{18225 \cdot 5}{5} - (\frac{5^{5}}{5} + 2025 \cdot 5) - 28 (\frac{9^{4}}{4} - \frac{5^{4}}{4}) + 286 (\frac{9^{3}}{3} - \frac{5^{3}}{3}) - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.5

Combine the numerators over the common denominator.

$V = π (\frac{59049 + 18225 \cdot 5}{5} - (\frac{5^{5}}{5} + 2025 \cdot 5) - 28 (\frac{9^{4}}{4} - \frac{5^{4}}{4}) + 286 (\frac{9^{3}}{3} - \frac{5^{3}}{3}) - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.6

Simplify the numerator.

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Step 16.2.5.6.1

Multiply $18225$ by $5$ .

$V = π (\frac{59049 + 91125}{5} - (\frac{5^{5}}{5} + 2025 \cdot 5) - 28 (\frac{9^{4}}{4} - \frac{5^{4}}{4}) + 286 (\frac{9^{3}}{3} - \frac{5^{3}}{3}) - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.6.2

Add $59049$ and $91125$ .

$V = π (\frac{150174}{5} - (\frac{5^{5}}{5} + 2025 \cdot 5) - 28 (\frac{9^{4}}{4} - \frac{5^{4}}{4}) + 286 (\frac{9^{3}}{3} - \frac{5^{3}}{3}) - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.7

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

$V = π (\frac{150174}{5} - (\frac{3125}{5} + 2025 \cdot 5) - 28 (\frac{9^{4}}{4} - \frac{5^{4}}{4}) + 286 (\frac{9^{3}}{3} - \frac{5^{3}}{3}) - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.8

Cancel the common factor of $3125$ and $5$ .

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Step 16.2.5.8.1

Factor $5$ out of $3125$ .

$V = π (\frac{150174}{5} - (\frac{5 \cdot 625}{5} + 2025 \cdot 5) - 28 (\frac{9^{4}}{4} - \frac{5^{4}}{4}) + 286 (\frac{9^{3}}{3} - \frac{5^{3}}{3}) - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.8.2

Cancel the common factors.

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Step 16.2.5.8.2.1

Factor $5$ out of $5$ .

$V = π (\frac{150174}{5} - (\frac{5 \cdot 625}{5 (1)} + 2025 \cdot 5) - 28 (\frac{9^{4}}{4} - \frac{5^{4}}{4}) + 286 (\frac{9^{3}}{3} - \frac{5^{3}}{3}) - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.8.2.2

Cancel the common factor.

$V = π (\frac{150174}{5} - (\frac{5 \cdot 625}{5 \cdot 1} + 2025 \cdot 5) - 28 (\frac{9^{4}}{4} - \frac{5^{4}}{4}) + 286 (\frac{9^{3}}{3} - \frac{5^{3}}{3}) - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.8.2.3

Rewrite the expression.

$V = π (\frac{150174}{5} - (\frac{625}{1} + 2025 \cdot 5) - 28 (\frac{9^{4}}{4} - \frac{5^{4}}{4}) + 286 (\frac{9^{3}}{3} - \frac{5^{3}}{3}) - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.8.2.4

Divide $625$ by $1$ .

$V = π (\frac{150174}{5} - (625 + 2025 \cdot 5) - 28 (\frac{9^{4}}{4} - \frac{5^{4}}{4}) + 286 (\frac{9^{3}}{3} - \frac{5^{3}}{3}) - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.9

Multiply $2025$ by $5$ .

$V = π (\frac{150174}{5} - (625 + 10125) - 28 (\frac{9^{4}}{4} - \frac{5^{4}}{4}) + 286 (\frac{9^{3}}{3} - \frac{5^{3}}{3}) - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.10

Add $625$ and $10125$ .

$V = π (\frac{150174}{5} - 1 \cdot 10750 - 28 (\frac{9^{4}}{4} - \frac{5^{4}}{4}) + 286 (\frac{9^{3}}{3} - \frac{5^{3}}{3}) - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.11

Multiply $- 1$ by $10750$ .

$V = π (\frac{150174}{5} - 10750 - 28 (\frac{9^{4}}{4} - \frac{5^{4}}{4}) + 286 (\frac{9^{3}}{3} - \frac{5^{3}}{3}) - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.12

To write $- 10750$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$V = π (\frac{150174}{5} - 10750 \cdot \frac{5}{5} - 28 (\frac{9^{4}}{4} - \frac{5^{4}}{4}) + 286 (\frac{9^{3}}{3} - \frac{5^{3}}{3}) - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.13

Combine $- 10750$ and $\frac{5}{5}$ .

$V = π (\frac{150174}{5} + \frac{- 10750 \cdot 5}{5} - 28 (\frac{9^{4}}{4} - \frac{5^{4}}{4}) + 286 (\frac{9^{3}}{3} - \frac{5^{3}}{3}) - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.14

Combine the numerators over the common denominator.

$V = π (\frac{150174 - 10750 \cdot 5}{5} - 28 (\frac{9^{4}}{4} - \frac{5^{4}}{4}) + 286 (\frac{9^{3}}{3} - \frac{5^{3}}{3}) - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.15

Simplify the numerator.

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Step 16.2.5.15.1

Multiply $- 10750$ by $5$ .

$V = π (\frac{150174 - 53750}{5} - 28 (\frac{9^{4}}{4} - \frac{5^{4}}{4}) + 286 (\frac{9^{3}}{3} - \frac{5^{3}}{3}) - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.15.2

Subtract $53750$ from $150174$ .

$V = π (\frac{96424}{5} - 28 (\frac{9^{4}}{4} - \frac{5^{4}}{4}) + 286 (\frac{9^{3}}{3} - \frac{5^{3}}{3}) - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.16

Raise $9$ to the power of $4$ .

$V = π (\frac{96424}{5} - 28 (\frac{6561}{4} - \frac{5^{4}}{4}) + 286 (\frac{9^{3}}{3} - \frac{5^{3}}{3}) - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.17

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

$V = π (\frac{96424}{5} - 28 (\frac{6561}{4} - \frac{625}{4}) + 286 (\frac{9^{3}}{3} - \frac{5^{3}}{3}) - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.18

Combine the numerators over the common denominator.

$V = π (\frac{96424}{5} - 28 \frac{6561 - 625}{4} + 286 (\frac{9^{3}}{3} - \frac{5^{3}}{3}) - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.19

Subtract $625$ from $6561$ .

$V = π (\frac{96424}{5} - 28 (\frac{5936}{4}) + 286 (\frac{9^{3}}{3} - \frac{5^{3}}{3}) - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.20

Cancel the common factor of $5936$ and $4$ .

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Step 16.2.5.20.1

Factor $4$ out of $5936$ .

$V = π (\frac{96424}{5} - 28 \frac{4 \cdot 1484}{4} + 286 (\frac{9^{3}}{3} - \frac{5^{3}}{3}) - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.20.2

Cancel the common factors.

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Step 16.2.5.20.2.1

Factor $4$ out of $4$ .

$V = π (\frac{96424}{5} - 28 \frac{4 \cdot 1484}{4 (1)} + 286 (\frac{9^{3}}{3} - \frac{5^{3}}{3}) - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.20.2.2

Cancel the common factor.

$V = π (\frac{96424}{5} - 28 \frac{4 \cdot 1484}{4 \cdot 1} + 286 (\frac{9^{3}}{3} - \frac{5^{3}}{3}) - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.20.2.3

Rewrite the expression.

$V = π (\frac{96424}{5} - 28 (\frac{1484}{1}) + 286 (\frac{9^{3}}{3} - \frac{5^{3}}{3}) - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.20.2.4

Divide $1484$ by $1$ .

$V = π (\frac{96424}{5} - 28 \cdot 1484 + 286 (\frac{9^{3}}{3} - \frac{5^{3}}{3}) - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.21

Multiply $- 28$ by $1484$ .

$V = π (\frac{96424}{5} - 41552 + 286 (\frac{9^{3}}{3} - \frac{5^{3}}{3}) - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.22

To write $- 41552$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$V = π (\frac{96424}{5} - 41552 \cdot \frac{5}{5} + 286 (\frac{9^{3}}{3} - \frac{5^{3}}{3}) - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.23

Combine $- 41552$ and $\frac{5}{5}$ .

$V = π (\frac{96424}{5} + \frac{- 41552 \cdot 5}{5} + 286 (\frac{9^{3}}{3} - \frac{5^{3}}{3}) - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.24

Combine the numerators over the common denominator.

$V = π (\frac{96424 - 41552 \cdot 5}{5} + 286 (\frac{9^{3}}{3} - \frac{5^{3}}{3}) - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.25

Simplify the numerator.

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Step 16.2.5.25.1

Multiply $- 41552$ by $5$ .

$V = π (\frac{96424 - 207760}{5} + 286 (\frac{9^{3}}{3} - \frac{5^{3}}{3}) - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.25.2

Subtract $207760$ from $96424$ .

$V = π (\frac{- 111336}{5} + 286 (\frac{9^{3}}{3} - \frac{5^{3}}{3}) - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.26

Move the negative in front of the fraction.

$V = π (- \frac{111336}{5} + 286 (\frac{9^{3}}{3} - \frac{5^{3}}{3}) - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.27

Raise $9$ to the power of $3$ .

$V = π (- \frac{111336}{5} + 286 (\frac{729}{3} - \frac{5^{3}}{3}) - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.28

Cancel the common factor of $729$ and $3$ .

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Step 16.2.5.28.1

Factor $3$ out of $729$ .

$V = π (- \frac{111336}{5} + 286 (\frac{3 \cdot 243}{3} - \frac{5^{3}}{3}) - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.28.2

Cancel the common factors.

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Step 16.2.5.28.2.1

Factor $3$ out of $3$ .

$V = π (- \frac{111336}{5} + 286 (\frac{3 \cdot 243}{3 (1)} - \frac{5^{3}}{3}) - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.28.2.2

Cancel the common factor.

$V = π (- \frac{111336}{5} + 286 (\frac{3 \cdot 243}{3 \cdot 1} - \frac{5^{3}}{3}) - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.28.2.3

Rewrite the expression.

$V = π (- \frac{111336}{5} + 286 (\frac{243}{1} - \frac{5^{3}}{3}) - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.28.2.4

Divide $243$ by $1$ .

$V = π (- \frac{111336}{5} + 286 (243 - \frac{5^{3}}{3}) - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.29

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

$V = π (- \frac{111336}{5} + 286 (243 - \frac{125}{3}) - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.30

To write $243$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$V = π (- \frac{111336}{5} + 286 (243 \cdot \frac{3}{3} - \frac{125}{3}) - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.31

Combine $243$ and $\frac{3}{3}$ .

$V = π (- \frac{111336}{5} + 286 (\frac{243 \cdot 3}{3} - \frac{125}{3}) - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.32

Combine the numerators over the common denominator.

$V = π (- \frac{111336}{5} + 286 (\frac{243 \cdot 3 - 125}{3}) - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.33

Simplify the numerator.

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Step 16.2.5.33.1

Multiply $243$ by $3$ .

$V = π (- \frac{111336}{5} + 286 (\frac{729 - 125}{3}) - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.33.2

Subtract $125$ from $729$ .

$V = π (- \frac{111336}{5} + 286 (\frac{604}{3}) - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.34

Combine $286$ and $\frac{604}{3}$ .

$V = π (- \frac{111336}{5} + \frac{286 \cdot 604}{3} - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.35

Multiply $286$ by $604$ .

$V = π (- \frac{111336}{5} + \frac{172744}{3} - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.36

To write $- \frac{111336}{5}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$V = π (- \frac{111336}{5} \cdot \frac{3}{3} + \frac{172744}{3} - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.37

To write $\frac{172744}{3}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$V = π (- \frac{111336}{5} \cdot \frac{3}{3} + \frac{172744}{3} \cdot \frac{5}{5} - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.38

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

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Step 16.2.5.38.1

Multiply $\frac{111336}{5}$ by $\frac{3}{3}$ .

$V = π (- \frac{111336 \cdot 3}{5 \cdot 3} + \frac{172744}{3} \cdot \frac{5}{5} - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.38.2

Multiply $5$ by $3$ .

$V = π (- \frac{111336 \cdot 3}{15} + \frac{172744}{3} \cdot \frac{5}{5} - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.38.3

Multiply $\frac{172744}{3}$ by $\frac{5}{5}$ .

$V = π (- \frac{111336 \cdot 3}{15} + \frac{172744 \cdot 5}{3 \cdot 5} - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.38.4

Multiply $3$ by $5$ .

$V = π (- \frac{111336 \cdot 3}{15} + \frac{172744 \cdot 5}{15} - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.39

Combine the numerators over the common denominator.

$V = π (\frac{- 111336 \cdot 3 + 172744 \cdot 5}{15} - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.40

Simplify the numerator.

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Step 16.2.5.40.1

Multiply $- 111336$ by $3$ .

$V = π (\frac{- 334008 + 172744 \cdot 5}{15} - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.40.2

Multiply $172744$ by $5$ .

$V = π (\frac{- 334008 + 863720}{15} - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.40.3

Add $- 334008$ and $863720$ .

$V = π (\frac{529712}{15} - 1260 ((\frac{9^{2}}{2}) - \frac{5^{2}}{2}))$

Step 16.2.5.41

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

$V = π (\frac{529712}{15} - 1260 (\frac{81}{2} - \frac{5^{2}}{2}))$

Step 16.2.5.42

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

$V = π (\frac{529712}{15} - 1260 (\frac{81}{2} - \frac{25}{2}))$

Step 16.2.5.43

Combine the numerators over the common denominator.

$V = π (\frac{529712}{15} - 1260 \frac{81 - 25}{2})$

Step 16.2.5.44

Subtract $25$ from $81$ .

$V = π (\frac{529712}{15} - 1260 (\frac{56}{2}))$

Step 16.2.5.45

Cancel the common factor of $56$ and $2$ .

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Step 16.2.5.45.1

Factor $2$ out of $56$ .

$V = π (\frac{529712}{15} - 1260 \frac{2 \cdot 28}{2})$

Step 16.2.5.45.2

Cancel the common factors.

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Step 16.2.5.45.2.1

Factor $2$ out of $2$ .

$V = π (\frac{529712}{15} - 1260 \frac{2 \cdot 28}{2 (1)})$

Step 16.2.5.45.2.2

Cancel the common factor.

$V = π (\frac{529712}{15} - 1260 \frac{2 \cdot 28}{2 \cdot 1})$

Step 16.2.5.45.2.3

Rewrite the expression.

$V = π (\frac{529712}{15} - 1260 (\frac{28}{1}))$

Step 16.2.5.45.2.4

Divide $28$ by $1$ .

$V = π (\frac{529712}{15} - 1260 \cdot 28)$

Step 16.2.5.46

Multiply $- 1260$ by $28$ .

$V = π (\frac{529712}{15} - 35280)$

Step 16.2.5.47

To write $- 35280$ as a fraction with a common denominator, multiply by $\frac{15}{15}$ .

$V = π (\frac{529712}{15} - 35280 \cdot \frac{15}{15})$

Step 16.2.5.48

Combine $- 35280$ and $\frac{15}{15}$ .

$V = π (\frac{529712}{15} + \frac{- 35280 \cdot 15}{15})$

Step 16.2.5.49

Combine the numerators over the common denominator.

$V = π (\frac{529712 - 35280 \cdot 15}{15})$

Step 16.2.5.50

Simplify the numerator.

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Step 16.2.5.50.1

Multiply $- 35280$ by $15$ .

$V = π (\frac{529712 - 529200}{15})$

Step 16.2.5.50.2

Subtract $529200$ from $529712$ .

$V = π (\frac{512}{15})$

Step 16.2.5.51

Combine $π$ and $\frac{512}{15}$ .

$V = \frac{π \cdot 512}{15}$

Step 16.2.5.52

Move $512$ to the left of $π$ .

$V = \frac{512 π}{15}$

Step 17

The result can be shown in multiple forms.

Exact Form:

$V = \frac{512 π}{15}$

Decimal Form:

$V = 107.23302924 \dots$

Step 18