Calculus Examples

y = - 7 x^{2} + 21 x

$y = - 7 x^{2} + 21 x$ ,

y = 3 x

$y = 3 x$

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)

$f (x)$ and

A = π r^{2}

$A = π r^{2}$ .

V = π \int_{0}^{2. ¯ ¯¯¯¯¯¯¯¯¯¯ ¯ 571428} {(f (x))}^{2} - {(g (x))}^{2} d x

$V = π \int_{0}^{2.\overline{571428}} {(f (x))}^{2} - {(g (x))}^{2} d x$ where

f (x) = - 7 x^{2} + 21 x

$f (x) = - 7 x^{2} + 21 x$ and

g (x) = 3 x

$g (x) = 3 x$

Step 2

Simplify the integrand.

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite

{(- 7 x^{2} + 21 x)}^{2}

${(- 7 x^{2} + 21 x)}^{2}$ as

(- 7 x^{2} + 21 x) (- 7 x^{2} + 21 x)

$(- 7 x^{2} + 21 x) (- 7 x^{2} + 21 x)$ .

V = (- 7 x^{2} + 21 x) (- 7 x^{2} + 21 x) - {(3 x)}^{2}

$V = (- 7 x^{2} + 21 x) (- 7 x^{2} + 21 x) - {(3 x)}^{2}$

Step 2.1.2

Expand

(- 7 x^{2} + 21 x) (- 7 x^{2} + 21 x)

$(- 7 x^{2} + 21 x) (- 7 x^{2} + 21 x)$ using the FOIL Method.

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Step 2.1.2.1

Apply the distributive property.

V = - 7 x^{2} (- 7 x^{2} + 21 x) + 21 x (- 7 x^{2} + 21 x) - {(3 x)}^{2}

$V = - 7 x^{2} (- 7 x^{2} + 21 x) + 21 x (- 7 x^{2} + 21 x) - {(3 x)}^{2}$

Step 2.1.2.2

Apply the distributive property.

V = - 7 x^{2} (- 7 x^{2}) - 7 x^{2} (21 x) + 21 x (- 7 x^{2} + 21 x) - {(3 x)}^{2}

$V = - 7 x^{2} (- 7 x^{2}) - 7 x^{2} (21 x) + 21 x (- 7 x^{2} + 21 x) - {(3 x)}^{2}$

Step 2.1.2.3

Apply the distributive property.

V = - 7 x^{2} (- 7 x^{2}) - 7 x^{2} (21 x) + 21 x (- 7 x^{2}) + 21 x (21 x) - {(3 x)}^{2}

$V = - 7 x^{2} (- 7 x^{2}) - 7 x^{2} (21 x) + 21 x (- 7 x^{2}) + 21 x (21 x) - {(3 x)}^{2}$

V = - 7 x^{2} (- 7 x^{2}) - 7 x^{2} (21 x) + 21 x (- 7 x^{2}) + 21 x (21 x) - {(3 x)}^{2}

$V = - 7 x^{2} (- 7 x^{2}) - 7 x^{2} (21 x) + 21 x (- 7 x^{2}) + 21 x (21 x) - {(3 x)}^{2}$

Step 2.1.3

Simplify and combine like terms.

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Step 2.1.3.1

Simplify each term.

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Step 2.1.3.1.1

Rewrite using the commutative property of multiplication.

V = - 7 \cdot (- 7 x^{2} x^{2}) - 7 x^{2} (21 x) + 21 x (- 7 x^{2}) + 21 x (21 x) - {(3 x)}^{2}

$V = - 7 \cdot (- 7 x^{2} x^{2}) - 7 x^{2} (21 x) + 21 x (- 7 x^{2}) + 21 x (21 x) - {(3 x)}^{2}$

Step 2.1.3.1.2

Multiply

x^{2}

$x^{2}$ by

x^{2}

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

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Step 2.1.3.1.2.1

Move

x^{2}

$x^{2}$ .

V = - 7 \cdot (- 7 (x^{2} x^{2})) - 7 x^{2} (21 x) + 21 x (- 7 x^{2}) + 21 x (21 x) - {(3 x)}^{2}

$V = - 7 \cdot (- 7 (x^{2} x^{2})) - 7 x^{2} (21 x) + 21 x (- 7 x^{2}) + 21 x (21 x) - {(3 x)}^{2}$

Step 2.1.3.1.2.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$V = - 7 \cdot (- 7 x^{2 + 2}) - 7 x^{2} (21 x) + 21 x (- 7 x^{2}) + 21 x (21 x) - {(3 x)}^{2}$

Step 2.1.3.1.2.3

Add

$2$ and

$2$ .

$V = - 7 \cdot (- 7 x^{4}) - 7 x^{2} (21 x) + 21 x (- 7 x^{2}) + 21 x (21 x) - {(3 x)}^{2}$

Step 2.1.3.1.3

Multiply

$- 7$ by

$- 7$ .

$V = 49 x^{4} - 7 x^{2} (21 x) + 21 x (- 7 x^{2}) + 21 x (21 x) - {(3 x)}^{2}$

Step 2.1.3.1.4

Rewrite using the commutative property of multiplication.

$V = 49 x^{4} - 7 \cdot (21 x^{2} x) + 21 x (- 7 x^{2}) + 21 x (21 x) - {(3 x)}^{2}$

Step 2.1.3.1.5

Multiply

$x^{2}$ by

$x$ by adding the exponents.

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Step 2.1.3.1.5.1

Move

$x$ .

$V = 49 x^{4} - 7 \cdot (21 (x \cdot x^{2})) + 21 x (- 7 x^{2}) + 21 x (21 x) - {(3 x)}^{2}$

Step 2.1.3.1.5.2

Multiply

$x$ by

$x^{2}$ .

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Step 2.1.3.1.5.2.1

Raise

$x$ to the power of

$1$ .

$V = 49 x^{4} - 7 \cdot (21 (x \cdot x^{2})) + 21 x (- 7 x^{2}) + 21 x (21 x) - {(3 x)}^{2}$

Step 2.1.3.1.5.2.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$V = 49 x^{4} - 7 \cdot (21 x^{1 + 2}) + 21 x (- 7 x^{2}) + 21 x (21 x) - {(3 x)}^{2}$

Step 2.1.3.1.5.3

Add

$1$ and

$2$ .

$V = 49 x^{4} - 7 \cdot (21 x^{3}) + 21 x (- 7 x^{2}) + 21 x (21 x) - {(3 x)}^{2}$

Step 2.1.3.1.6

Multiply

$- 7$ by

$21$ .

$V = 49 x^{4} - 147 x^{3} + 21 x (- 7 x^{2}) + 21 x (21 x) - {(3 x)}^{2}$

Step 2.1.3.1.7

Rewrite using the commutative property of multiplication.

$V = 49 x^{4} - 147 x^{3} + 21 \cdot (- 7 x \cdot x^{2}) + 21 x (21 x) - {(3 x)}^{2}$

Step 2.1.3.1.8

Multiply

$x$ by

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

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Step 2.1.3.1.8.1

Move

$x^{2}$ .

$V = 49 x^{4} - 147 x^{3} + 21 \cdot (- 7 (x^{2} x)) + 21 x (21 x) - {(3 x)}^{2}$

Step 2.1.3.1.8.2

Multiply

$x^{2}$ by

$x$ .

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Step 2.1.3.1.8.2.1

Raise

$x$ to the power of

$1$ .

$V = 49 x^{4} - 147 x^{3} + 21 \cdot (- 7 (x^{2} x)) + 21 x (21 x) - {(3 x)}^{2}$

Step 2.1.3.1.8.2.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$V = 49 x^{4} - 147 x^{3} + 21 \cdot (- 7 x^{2 + 1}) + 21 x (21 x) - {(3 x)}^{2}$

Step 2.1.3.1.8.3

Add

$2$ and

$1$ .

$V = 49 x^{4} - 147 x^{3} + 21 \cdot (- 7 x^{3}) + 21 x (21 x) - {(3 x)}^{2}$

Step 2.1.3.1.9

Multiply

$21$ by

$- 7$ .

$V = 49 x^{4} - 147 x^{3} - 147 x^{3} + 21 x (21 x) - {(3 x)}^{2}$

Step 2.1.3.1.10

Rewrite using the commutative property of multiplication.

$V = 49 x^{4} - 147 x^{3} - 147 x^{3} + 21 \cdot (21 x \cdot x) - {(3 x)}^{2}$

Step 2.1.3.1.11

Multiply

$x$ by

$x$ by adding the exponents.

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Step 2.1.3.1.11.1

Move

$x$ .

$V = 49 x^{4} - 147 x^{3} - 147 x^{3} + 21 \cdot (21 (x \cdot x)) - {(3 x)}^{2}$

Step 2.1.3.1.11.2

Multiply

$x$ by

$x$ .

$V = 49 x^{4} - 147 x^{3} - 147 x^{3} + 21 \cdot (21 x^{2}) - {(3 x)}^{2}$

Step 2.1.3.1.12

Multiply

$21$ by

$21$ .

$V = 49 x^{4} - 147 x^{3} - 147 x^{3} + 441 x^{2} - {(3 x)}^{2}$

Step 2.1.3.2

Subtract

$147 x^{3}$ from

$- 147 x^{3}$ .

$V = 49 x^{4} - 294 x^{3} + 441 x^{2} - {(3 x)}^{2}$

Step 2.1.4

Apply the product rule to

$3 x$ .

$V = 49 x^{4} - 294 x^{3} + 441 x^{2} - (3^{2} x^{2})$

Step 2.1.5

Raise

$3$ to the power of

$2$ .

$V = 49 x^{4} - 294 x^{3} + 441 x^{2} - (9 x^{2})$

Step 2.1.6

Multiply

$9$ by

$- 1$ .

$V = 49 x^{4} - 294 x^{3} + 441 x^{2} - 9 x^{2}$

Step 2.2

Subtract

$9 x^{2}$ from

$441 x^{2}$ .

$V = 49 x^{4} - 294 x^{3} + 432 x^{2}$

Step 3

Split the single integral into multiple integrals.

$V = π (\int_{0}^{2.\overline{571428}} 49 x^{4} d x + \int_{0}^{2.\overline{571428}} - 294 x^{3} d x + \int_{0}^{2.\overline{571428}} 432 x^{2} d x)$

Step 4

Since

$49$ is constant with respect to

$x$ , move

$49$ out of the integral.

$V = π (49 \int_{0}^{2.\overline{571428}} x^{4} d x + \int_{0}^{2.\overline{571428}} - 294 x^{3} d x + \int_{0}^{2.\overline{571428}} 432 x^{2} d x)$

Step 5

By the Power Rule, the integral of

$x^{4}$ with respect to

$x$ is

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

$V = π (49 (\frac{1}{5} x^{5}]_{0}^{2.\overline{571428}}) + \int_{0}^{2.\overline{571428}} - 294 x^{3} d x + \int_{0}^{2.\overline{571428}} 432 x^{2} d x)$

Step 6

Combine

$\frac{1}{5}$ and

$x^{5}$ .

$V = π (49 (\frac{x^{5}}{5}]_{0}^{2.\overline{571428}}) + \int_{0}^{2.\overline{571428}} - 294 x^{3} d x + \int_{0}^{2.\overline{571428}} 432 x^{2} d x)$

Step 7

Since

$- 294$ is constant with respect to

$x$ , move

$- 294$ out of the integral.

$V = π (49 (\frac{x^{5}}{5}]_{0}^{2.\overline{571428}}) - 294 \int_{0}^{2.\overline{571428}} x^{3} d x + \int_{0}^{2.\overline{571428}} 432 x^{2} d x)$

Step 8

By the Power Rule, the integral of

$x^{3}$ with respect to

$x$ is

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

$V = π (49 (\frac{x^{5}}{5}]_{0}^{2.\overline{571428}}) - 294 (\frac{1}{4} x^{4}]_{0}^{2.\overline{571428}}) + \int_{0}^{2.\overline{571428}} 432 x^{2} d x)$

Step 9

Combine

$\frac{1}{4}$ and

$x^{4}$ .

$V = π (49 (\frac{x^{5}}{5}]_{0}^{2.\overline{571428}}) - 294 (\frac{x^{4}}{4}]_{0}^{2.\overline{571428}}) + \int_{0}^{2.\overline{571428}} 432 x^{2} d x)$

Step 10

Since

$432$ is constant with respect to

$x$ , move

$432$ out of the integral.

$V = π (49 (\frac{x^{5}}{5}]_{0}^{2.\overline{571428}}) - 294 (\frac{x^{4}}{4}]_{0}^{2.\overline{571428}}) + 432 \int_{0}^{2.\overline{571428}} x^{2} d x)$

Step 11

By the Power Rule, the integral of

$x^{2}$ with respect to

$x$ is

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

$V = π (49 (\frac{x^{5}}{5}]_{0}^{2.\overline{571428}}) - 294 (\frac{x^{4}}{4}]_{0}^{2.\overline{571428}}) + 432 (\frac{1}{3} x^{3}]_{0}^{2.\overline{571428}}))$

Step 12

Simplify the answer.

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Step 12.1

Combine

$\frac{1}{3}$ and

$x^{3}$ .

$V = π (49 (\frac{x^{5}}{5}]_{0}^{2.\overline{571428}}) - 294 (\frac{x^{4}}{4}]_{0}^{2.\overline{571428}}) + 432 (\frac{x^{3}}{3}]_{0}^{2.\overline{571428}}))$

Step 12.2

Substitute and simplify.

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Step 12.2.1

Evaluate

$\frac{x^{5}}{5}$ at

$2.\overline{571428}$ and at

$0$ .

$V = π (49 ((\frac{{2.\overline{571428}}^{5}}{5}) - \frac{0^{5}}{5}) - 294 (\frac{x^{4}}{4}]_{0}^{2.\overline{571428}}) + 432 (\frac{x^{3}}{3}]_{0}^{2.\overline{571428}}))$

Step 12.2.2

Evaluate

$\frac{x^{4}}{4}$ at

$2.\overline{571428}$ and at

$0$ .

$V = π (49 (\frac{{2.\overline{571428}}^{5}}{5} - \frac{0^{5}}{5}) - 294 (\frac{{2.\overline{571428}}^{4}}{4} - \frac{0^{4}}{4}) + 432 (\frac{x^{3}}{3}]_{0}^{2.\overline{571428}}))$

Step 12.2.3

Evaluate

$\frac{x^{3}}{3}$ at

$2.\overline{571428}$ and at

$0$ .

$V = π (49 (\frac{{2.\overline{571428}}^{5}}{5} - \frac{0^{5}}{5}) - 294 (\frac{{2.\overline{571428}}^{4}}{4} - \frac{0^{4}}{4}) + 432 ((\frac{{2.\overline{571428}}^{3}}{3}) - \frac{0^{3}}{3}))$

Step 12.2.4

Simplify.

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Step 12.2.4.1

Raise

$2.\overline{571428}$ to the power of

$5$ .

$V = π (49 (\frac{112.42744094}{5} - \frac{0^{5}}{5}) - 294 (\frac{{2.\overline{571428}}^{4}}{4} - \frac{0^{4}}{4}) + 432 ((\frac{{2.\overline{571428}}^{3}}{3}) - \frac{0^{3}}{3}))$

Step 12.2.4.2

Raising

$0$ to any positive power yields

$0$ .

$V = π (49 (\frac{112.42744094}{5} - \frac{0}{5}) - 294 (\frac{{2.\overline{571428}}^{4}}{4} - \frac{0^{4}}{4}) + 432 ((\frac{{2.\overline{571428}}^{3}}{3}) - \frac{0^{3}}{3}))$

Step 12.2.4.3

Cancel the common factor of

$0$ and

$5$ .

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Step 12.2.4.3.1

Factor

$5$ out of

$0$ .

$V = π (49 (\frac{112.42744094}{5} - \frac{5 (0)}{5}) - 294 (\frac{{2.\overline{571428}}^{4}}{4} - \frac{0^{4}}{4}) + 432 ((\frac{{2.\overline{571428}}^{3}}{3}) - \frac{0^{3}}{3}))$

Step 12.2.4.3.2

Cancel the common factors.

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Step 12.2.4.3.2.1

Factor

$5$ out of

$5$ .

$V = π (49 (\frac{112.42744094}{5} - \frac{5 \cdot 0}{5 \cdot 1}) - 294 (\frac{{2.\overline{571428}}^{4}}{4} - \frac{0^{4}}{4}) + 432 ((\frac{{2.\overline{571428}}^{3}}{3}) - \frac{0^{3}}{3}))$

Step 12.2.4.3.2.2

Cancel the common factor.

$V = π (49 (\frac{112.42744094}{5} - \frac{5 \cdot 0}{5 \cdot 1}) - 294 (\frac{{2.\overline{571428}}^{4}}{4} - \frac{0^{4}}{4}) + 432 ((\frac{{2.\overline{571428}}^{3}}{3}) - \frac{0^{3}}{3}))$

Step 12.2.4.3.2.3

Rewrite the expression.

$V = π (49 (\frac{112.42744094}{5} - \frac{0}{1}) - 294 (\frac{{2.\overline{571428}}^{4}}{4} - \frac{0^{4}}{4}) + 432 ((\frac{{2.\overline{571428}}^{3}}{3}) - \frac{0^{3}}{3}))$

Step 12.2.4.3.2.4

Divide

$0$ by

$1$ .

$V = π (49 (\frac{112.42744094}{5} - 0) - 294 (\frac{{2.\overline{571428}}^{4}}{4} - \frac{0^{4}}{4}) + 432 ((\frac{{2.\overline{571428}}^{3}}{3}) - \frac{0^{3}}{3}))$

Step 12.2.4.4

Multiply

$- 1$ by

$0$ .

$V = π (49 (\frac{112.42744094}{5} + 0) - 294 (\frac{{2.\overline{571428}}^{4}}{4} - \frac{0^{4}}{4}) + 432 ((\frac{{2.\overline{571428}}^{3}}{3}) - \frac{0^{3}}{3}))$

Step 12.2.4.5

Add

$\frac{112.42744094}{5}$ and

$0$ .

$V = π (49 (\frac{112.42744094}{5}) - 294 (\frac{{2.\overline{571428}}^{4}}{4} - \frac{0^{4}}{4}) + 432 ((\frac{{2.\overline{571428}}^{3}}{3}) - \frac{0^{3}}{3}))$

Step 12.2.4.6

Combine

$49$ and

$\frac{112.42744094}{5}$ .

$V = π (\frac{49 \cdot 112.42744094}{5} - 294 (\frac{{2.\overline{571428}}^{4}}{4} - \frac{0^{4}}{4}) + 432 ((\frac{{2.\overline{571428}}^{3}}{3}) - \frac{0^{3}}{3}))$

Step 12.2.4.7

Multiply

$49$ by

$112.42744094$ .

$V = π (\frac{5508.94460641}{5} - 294 (\frac{{2.\overline{571428}}^{4}}{4} - \frac{0^{4}}{4}) + 432 ((\frac{{2.\overline{571428}}^{3}}{3}) - \frac{0^{3}}{3}))$

Step 12.2.4.8

Raise

$2.\overline{571428}$ to the power of

$4$ .

$V = π (\frac{5508.94460641}{5} - 294 (\frac{43.72178259}{4} - \frac{0^{4}}{4}) + 432 ((\frac{{2.\overline{571428}}^{3}}{3}) - \frac{0^{3}}{3}))$

Step 12.2.4.9

Raising

$0$ to any positive power yields

$0$ .

$V = π (\frac{5508.94460641}{5} - 294 (\frac{43.72178259}{4} - \frac{0}{4}) + 432 ((\frac{{2.\overline{571428}}^{3}}{3}) - \frac{0^{3}}{3}))$

Step 12.2.4.10

Cancel the common factor of

$0$ and

$4$ .

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Step 12.2.4.10.1

Factor

$4$ out of

$0$ .

$V = π (\frac{5508.94460641}{5} - 294 (\frac{43.72178259}{4} - \frac{4 (0)}{4}) + 432 ((\frac{{2.\overline{571428}}^{3}}{3}) - \frac{0^{3}}{3}))$

Step 12.2.4.10.2

Cancel the common factors.

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Step 12.2.4.10.2.1

Factor

$4$ out of

$4$ .

$V = π (\frac{5508.94460641}{5} - 294 (\frac{43.72178259}{4} - \frac{4 \cdot 0}{4 \cdot 1}) + 432 ((\frac{{2.\overline{571428}}^{3}}{3}) - \frac{0^{3}}{3}))$

Step 12.2.4.10.2.2

Cancel the common factor.

$V = π (\frac{5508.94460641}{5} - 294 (\frac{43.72178259}{4} - \frac{4 \cdot 0}{4 \cdot 1}) + 432 ((\frac{{2.\overline{571428}}^{3}}{3}) - \frac{0^{3}}{3}))$

Step 12.2.4.10.2.3

Rewrite the expression.

$V = π (\frac{5508.94460641}{5} - 294 (\frac{43.72178259}{4} - \frac{0}{1}) + 432 ((\frac{{2.\overline{571428}}^{3}}{3}) - \frac{0^{3}}{3}))$

Step 12.2.4.10.2.4

Divide

$0$ by

$1$ .

$V = π (\frac{5508.94460641}{5} - 294 (\frac{43.72178259}{4} - 0) + 432 ((\frac{{2.\overline{571428}}^{3}}{3}) - \frac{0^{3}}{3}))$

Step 12.2.4.11

Multiply

$- 1$ by

$0$ .

$V = π (\frac{5508.94460641}{5} - 294 (\frac{43.72178259}{4} + 0) + 432 ((\frac{{2.\overline{571428}}^{3}}{3}) - \frac{0^{3}}{3}))$

Step 12.2.4.12

Add

$\frac{43.72178259}{4}$ and

$0$ .

$V = π (\frac{5508.94460641}{5} - 294 (\frac{43.72178259}{4}) + 432 ((\frac{{2.\overline{571428}}^{3}}{3}) - \frac{0^{3}}{3}))$

Step 12.2.4.13

Combine

$- 294$ and

$\frac{43.72178259}{4}$ .

$V = π (\frac{5508.94460641}{5} + \frac{- 294 \cdot 43.72178259}{4} + 432 ((\frac{{2.\overline{571428}}^{3}}{3}) - \frac{0^{3}}{3}))$

Step 12.2.4.14

Multiply

$- 294$ by

$43.72178259$ .

$V = π (\frac{5508.94460641}{5} + \frac{- 12854.20408163}{4} + 432 ((\frac{{2.\overline{571428}}^{3}}{3}) - \frac{0^{3}}{3}))$

Step 12.2.4.15

Move the negative in front of the fraction.

$V = π (\frac{5508.94460641}{5} - \frac{12854.20408163}{4} + 432 ((\frac{{2.\overline{571428}}^{3}}{3}) - \frac{0^{3}}{3}))$

Step 12.2.4.16

To write

$\frac{5508.94460641}{5}$ as a fraction with a common denominator, multiply by

$\frac{4}{4}$ .

$V = π (\frac{5508.94460641}{5} \cdot \frac{4}{4} - \frac{12854.20408163}{4} + 432 ((\frac{{2.\overline{571428}}^{3}}{3}) - \frac{0^{3}}{3}))$

Step 12.2.4.17

To write

$- \frac{12854.20408163}{4}$ as a fraction with a common denominator, multiply by

$\frac{5}{5}$ .

$V = π (\frac{5508.94460641}{5} \cdot \frac{4}{4} - \frac{12854.20408163}{4} \cdot \frac{5}{5} + 432 ((\frac{{2.\overline{571428}}^{3}}{3}) - \frac{0^{3}}{3}))$

Step 12.2.4.18

Write each expression with a common denominator of

$20$ , by multiplying each by an appropriate factor of

$1$ .

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Step 12.2.4.18.1

Multiply

$\frac{5508.94460641}{5}$ by

$\frac{4}{4}$ .

$V = π (\frac{5508.94460641 \cdot 4}{5 \cdot 4} - \frac{12854.20408163}{4} \cdot \frac{5}{5} + 432 ((\frac{{2.\overline{571428}}^{3}}{3}) - \frac{0^{3}}{3}))$

Step 12.2.4.18.2

Multiply

$5$ by

$4$ .

$V = π (\frac{5508.94460641 \cdot 4}{20} - \frac{12854.20408163}{4} \cdot \frac{5}{5} + 432 ((\frac{{2.\overline{571428}}^{3}}{3}) - \frac{0^{3}}{3}))$

Step 12.2.4.18.3

Multiply

$\frac{12854.20408163}{4}$ by

$\frac{5}{5}$ .

$V = π (\frac{5508.94460641 \cdot 4}{20} - \frac{12854.20408163 \cdot 5}{4 \cdot 5} + 432 ((\frac{{2.\overline{571428}}^{3}}{3}) - \frac{0^{3}}{3}))$

Step 12.2.4.18.4

Multiply

$4$ by

$5$ .

$V = π (\frac{5508.94460641 \cdot 4}{20} - \frac{12854.20408163 \cdot 5}{20} + 432 ((\frac{{2.\overline{571428}}^{3}}{3}) - \frac{0^{3}}{3}))$

Step 12.2.4.19

Combine the numerators over the common denominator.

$V = π (\frac{5508.94460641 \cdot 4 - 12854.20408163 \cdot 5}{20} + 432 ((\frac{{2.\overline{571428}}^{3}}{3}) - \frac{0^{3}}{3}))$

Step 12.2.4.20

Multiply

$5508.94460641$ by

$4$ .

$V = π (\frac{22035.77842565 - 12854.20408163 \cdot 5}{20} + 432 ((\frac{{2.\overline{571428}}^{3}}{3}) - \frac{0^{3}}{3}))$

Step 12.2.4.21

Multiply

$- 12854.20408163$ by

$5$ .

$V = π (\frac{22035.77842565 - 64271.02040816}{20} + 432 ((\frac{{2.\overline{571428}}^{3}}{3}) - \frac{0^{3}}{3}))$

Step 12.2.4.22

Subtract

$64271.02040816$ from

$22035.77842565$ .

$V = π (\frac{- 42235.2419825}{20} + 432 ((\frac{{2.\overline{571428}}^{3}}{3}) - \frac{0^{3}}{3}))$

Step 12.2.4.23

Move the negative in front of the fraction.

$V = π (- \frac{42235.2419825}{20} + 432 ((\frac{{2.\overline{571428}}^{3}}{3}) - \frac{0^{3}}{3}))$

Step 12.2.4.24

Raise

$2.\overline{571428}$ to the power of

$3$ .

$V = π (- \frac{42235.2419825}{20} + 432 (\frac{17.00291545}{3} - \frac{0^{3}}{3}))$

Step 12.2.4.25

Raising

$0$ to any positive power yields

$0$ .

$V = π (- \frac{42235.2419825}{20} + 432 (\frac{17.00291545}{3} - \frac{0}{3}))$

Step 12.2.4.26

Cancel the common factor of

$0$ and

$3$ .

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Step 12.2.4.26.1

Factor

$3$ out of

$0$ .

$V = π (- \frac{42235.2419825}{20} + 432 (\frac{17.00291545}{3} - \frac{3 (0)}{3}))$

Step 12.2.4.26.2

Cancel the common factors.

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Step 12.2.4.26.2.1

Factor

$3$ out of

$3$ .

$V = π (- \frac{42235.2419825}{20} + 432 (\frac{17.00291545}{3} - \frac{3 \cdot 0}{3 \cdot 1}))$

Step 12.2.4.26.2.2

Cancel the common factor.

$V = π (- \frac{42235.2419825}{20} + 432 (\frac{17.00291545}{3} - \frac{3 \cdot 0}{3 \cdot 1}))$

Step 12.2.4.26.2.3

Rewrite the expression.

$V = π (- \frac{42235.2419825}{20} + 432 (\frac{17.00291545}{3} - \frac{0}{1}))$

Step 12.2.4.26.2.4

Divide

$0$ by

$1$ .

$V = π (- \frac{42235.2419825}{20} + 432 (\frac{17.00291545}{3} - 0))$

Step 12.2.4.27

Multiply

$- 1$ by

$0$ .

$V = π (- \frac{42235.2419825}{20} + 432 (\frac{17.00291545}{3} + 0))$

Step 12.2.4.28

Add

$\frac{17.00291545}{3}$ and

$0$ .

$V = π (- \frac{42235.2419825}{20} + 432 (\frac{17.00291545}{3}))$

Step 12.2.4.29

Combine

$432$ and

$\frac{17.00291545}{3}$ .

$V = π (- \frac{42235.2419825}{20} + \frac{432 \cdot 17.00291545}{3})$

Step 12.2.4.30

Multiply

$432$ by

$17.00291545$ .

$V = π (- \frac{42235.2419825}{20} + \frac{7345.25947521}{3})$

Step 12.2.4.31

To write

$- \frac{42235.2419825}{20}$ as a fraction with a common denominator, multiply by

$\frac{3}{3}$ .

$V = π (- \frac{42235.2419825}{20} \cdot \frac{3}{3} + \frac{7345.25947521}{3})$

Step 12.2.4.32

To write

$\frac{7345.25947521}{3}$ as a fraction with a common denominator, multiply by

$\frac{20}{20}$ .

$V = π (- \frac{42235.2419825}{20} \cdot \frac{3}{3} + \frac{7345.25947521}{3} \cdot \frac{20}{20})$

Step 12.2.4.33

Write each expression with a common denominator of

$60$ , by multiplying each by an appropriate factor of

$1$ .

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Step 12.2.4.33.1

Multiply

$\frac{42235.2419825}{20}$ by

$\frac{3}{3}$ .

$V = π (- \frac{42235.2419825 \cdot 3}{20 \cdot 3} + \frac{7345.25947521}{3} \cdot \frac{20}{20})$

Step 12.2.4.33.2

Multiply

$20$ by

$3$ .

$V = π (- \frac{42235.2419825 \cdot 3}{60} + \frac{7345.25947521}{3} \cdot \frac{20}{20})$

Step 12.2.4.33.3

Multiply

$\frac{7345.25947521}{3}$ by

$\frac{20}{20}$ .

$V = π (- \frac{42235.2419825 \cdot 3}{60} + \frac{7345.25947521 \cdot 20}{3 \cdot 20})$

Step 12.2.4.33.4

Multiply

$3$ by

$20$ .

$V = π (- \frac{42235.2419825 \cdot 3}{60} + \frac{7345.25947521 \cdot 20}{60})$

Step 12.2.4.34

Combine the numerators over the common denominator.

$V = π (\frac{- 42235.2419825 \cdot 3 + 7345.25947521 \cdot 20}{60})$

Step 12.2.4.35

Multiply

$- 42235.2419825$ by

$3$ .

$V = π (\frac{- 126705.72594752 + 7345.25947521 \cdot 20}{60})$

Step 12.2.4.36

Multiply

$7345.25947521$ by

$20$ .

$V = π (\frac{- 126705.72594752 + 146905.18950437}{60})$

Step 12.2.4.37

Add

$- 126705.72594752$ and

$146905.18950437$ .

$V = π (\frac{20199.46355685}{60})$

Step 12.2.4.38

Combine

$π$ and

$\frac{20199.46355685}{60}$ .

$V = \frac{π \cdot 20199.46355685}{60}$

Step 12.2.4.39

Multiply

$π$ by

$20199.46355685$ .

$V = \frac{63458.48631665}{60}$

Step 13

Divide

$63458.48631665$ by

$60$ .

$V = 1057.64143861$

Step 14

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