Calculus Examples

$y = x^{3} - 10 x^{2} + 25 x$ , $y = x$

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 $f (x)$ and $A = π r^{2}$ .

$V = π \int_{0}^{4} {(f (x))}^{2} - {(g (x))}^{2} d x$ where $f (x) = x^{3} - 10 x^{2} + 25 x$ and $g (x) = x$

Step 2

Simplify the integrand.

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite ${(x^{3} - 10 x^{2} + 25 x)}^{2}$ as $(x^{3} - 10 x^{2} + 25 x) (x^{3} - 10 x^{2} + 25 x)$ .

$V = (x^{3} - 10 x^{2} + 25 x) (x^{3} - 10 x^{2} + 25 x) - {(x)}^{2}$

Step 2.1.2

Expand $(x^{3} - 10 x^{2} + 25 x) (x^{3} - 10 x^{2} + 25 x)$ by multiplying each term in the first expression by each term in the second expression.

$V = x^{3} x^{3} + x^{3} (- 10 x^{2}) + x^{3} (25 x) - 10 x^{2} x^{3} - 10 x^{2} (- 10 x^{2}) - 10 x^{2} (25 x) + 25 x \cdot x^{3} + 25 x (- 10 x^{2}) + 25 x (25 x) - {(x)}^{2}$

Step 2.1.3

Simplify each term.

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Step 2.1.3.1

Multiply $x^{3}$ by $x^{3}$ by adding the exponents.

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Step 2.1.3.1.1

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

$V = x^{3 + 3} + x^{3} (- 10 x^{2}) + x^{3} (25 x) - 10 x^{2} x^{3} - 10 x^{2} (- 10 x^{2}) - 10 x^{2} (25 x) + 25 x \cdot x^{3} + 25 x (- 10 x^{2}) + 25 x (25 x) - {(x)}^{2}$

Step 2.1.3.1.2

Add $3$ and $3$ .

$V = x^{6} + x^{3} (- 10 x^{2}) + x^{3} (25 x) - 10 x^{2} x^{3} - 10 x^{2} (- 10 x^{2}) - 10 x^{2} (25 x) + 25 x \cdot x^{3} + 25 x (- 10 x^{2}) + 25 x (25 x) - {(x)}^{2}$

Step 2.1.3.2

Rewrite using the commutative property of multiplication.

$V = x^{6} - 10 x^{3} x^{2} + x^{3} (25 x) - 10 x^{2} x^{3} - 10 x^{2} (- 10 x^{2}) - 10 x^{2} (25 x) + 25 x \cdot x^{3} + 25 x (- 10 x^{2}) + 25 x (25 x) - {(x)}^{2}$

Step 2.1.3.3

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

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Step 2.1.3.3.1

Move $x^{2}$ .

$V = x^{6} - 10 (x^{2} x^{3}) + x^{3} (25 x) - 10 x^{2} x^{3} - 10 x^{2} (- 10 x^{2}) - 10 x^{2} (25 x) + 25 x \cdot x^{3} + 25 x (- 10 x^{2}) + 25 x (25 x) - {(x)}^{2}$

Step 2.1.3.3.2

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

$V = x^{6} - 10 x^{2 + 3} + x^{3} (25 x) - 10 x^{2} x^{3} - 10 x^{2} (- 10 x^{2}) - 10 x^{2} (25 x) + 25 x \cdot x^{3} + 25 x (- 10 x^{2}) + 25 x (25 x) - {(x)}^{2}$

Step 2.1.3.3.3

Add $2$ and $3$ .

$V = x^{6} - 10 x^{5} + x^{3} (25 x) - 10 x^{2} x^{3} - 10 x^{2} (- 10 x^{2}) - 10 x^{2} (25 x) + 25 x \cdot x^{3} + 25 x (- 10 x^{2}) + 25 x (25 x) - {(x)}^{2}$

Step 2.1.3.4

Rewrite using the commutative property of multiplication.

$V = x^{6} - 10 x^{5} + 25 x^{3} x - 10 x^{2} x^{3} - 10 x^{2} (- 10 x^{2}) - 10 x^{2} (25 x) + 25 x \cdot x^{3} + 25 x (- 10 x^{2}) + 25 x (25 x) - {(x)}^{2}$

Step 2.1.3.5

Multiply $x^{3}$ by $x$ by adding the exponents.

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Step 2.1.3.5.1

Move $x$ .

$V = x^{6} - 10 x^{5} + 25 (x \cdot x^{3}) - 10 x^{2} x^{3} - 10 x^{2} (- 10 x^{2}) - 10 x^{2} (25 x) + 25 x \cdot x^{3} + 25 x (- 10 x^{2}) + 25 x (25 x) - {(x)}^{2}$

Step 2.1.3.5.2

Multiply $x$ by $x^{3}$ .

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Step 2.1.3.5.2.1

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

$V = x^{6} - 10 x^{5} + 25 (x \cdot x^{3}) - 10 x^{2} x^{3} - 10 x^{2} (- 10 x^{2}) - 10 x^{2} (25 x) + 25 x \cdot x^{3} + 25 x (- 10 x^{2}) + 25 x (25 x) - {(x)}^{2}$

Step 2.1.3.5.2.2

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

$V = x^{6} - 10 x^{5} + 25 x^{1 + 3} - 10 x^{2} x^{3} - 10 x^{2} (- 10 x^{2}) - 10 x^{2} (25 x) + 25 x \cdot x^{3} + 25 x (- 10 x^{2}) + 25 x (25 x) - {(x)}^{2}$

Step 2.1.3.5.3

Add $1$ and $3$ .

$V = x^{6} - 10 x^{5} + 25 x^{4} - 10 x^{2} x^{3} - 10 x^{2} (- 10 x^{2}) - 10 x^{2} (25 x) + 25 x \cdot x^{3} + 25 x (- 10 x^{2}) + 25 x (25 x) - {(x)}^{2}$

Step 2.1.3.6

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

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Step 2.1.3.6.1

Move $x^{3}$ .

$V = x^{6} - 10 x^{5} + 25 x^{4} - 10 (x^{3} x^{2}) - 10 x^{2} (- 10 x^{2}) - 10 x^{2} (25 x) + 25 x \cdot x^{3} + 25 x (- 10 x^{2}) + 25 x (25 x) - {(x)}^{2}$

Step 2.1.3.6.2

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

$V = x^{6} - 10 x^{5} + 25 x^{4} - 10 x^{3 + 2} - 10 x^{2} (- 10 x^{2}) - 10 x^{2} (25 x) + 25 x \cdot x^{3} + 25 x (- 10 x^{2}) + 25 x (25 x) - {(x)}^{2}$

Step 2.1.3.6.3

Add $3$ and $2$ .

$V = x^{6} - 10 x^{5} + 25 x^{4} - 10 x^{5} - 10 x^{2} (- 10 x^{2}) - 10 x^{2} (25 x) + 25 x \cdot x^{3} + 25 x (- 10 x^{2}) + 25 x (25 x) - {(x)}^{2}$

Step 2.1.3.7

Rewrite using the commutative property of multiplication.

$V = x^{6} - 10 x^{5} + 25 x^{4} - 10 x^{5} - 10 \cdot (- 10 x^{2} x^{2}) - 10 x^{2} (25 x) + 25 x \cdot x^{3} + 25 x (- 10 x^{2}) + 25 x (25 x) - {(x)}^{2}$

Step 2.1.3.8

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

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Step 2.1.3.8.1

Move $x^{2}$ .

$V = x^{6} - 10 x^{5} + 25 x^{4} - 10 x^{5} - 10 \cdot (- 10 (x^{2} x^{2})) - 10 x^{2} (25 x) + 25 x \cdot x^{3} + 25 x (- 10 x^{2}) + 25 x (25 x) - {(x)}^{2}$

Step 2.1.3.8.2

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

$V = x^{6} - 10 x^{5} + 25 x^{4} - 10 x^{5} - 10 \cdot (- 10 x^{2 + 2}) - 10 x^{2} (25 x) + 25 x \cdot x^{3} + 25 x (- 10 x^{2}) + 25 x (25 x) - {(x)}^{2}$

Step 2.1.3.8.3

Add $2$ and $2$ .

$V = x^{6} - 10 x^{5} + 25 x^{4} - 10 x^{5} - 10 \cdot (- 10 x^{4}) - 10 x^{2} (25 x) + 25 x \cdot x^{3} + 25 x (- 10 x^{2}) + 25 x (25 x) - {(x)}^{2}$

Step 2.1.3.9

Multiply $- 10$ by $- 10$ .

$V = x^{6} - 10 x^{5} + 25 x^{4} - 10 x^{5} + 100 x^{4} - 10 x^{2} (25 x) + 25 x \cdot x^{3} + 25 x (- 10 x^{2}) + 25 x (25 x) - {(x)}^{2}$

Step 2.1.3.10

Rewrite using the commutative property of multiplication.

$V = x^{6} - 10 x^{5} + 25 x^{4} - 10 x^{5} + 100 x^{4} - 10 \cdot (25 x^{2} x) + 25 x \cdot x^{3} + 25 x (- 10 x^{2}) + 25 x (25 x) - {(x)}^{2}$

Step 2.1.3.11

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

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Step 2.1.3.11.1

Move $x$ .

$V = x^{6} - 10 x^{5} + 25 x^{4} - 10 x^{5} + 100 x^{4} - 10 \cdot (25 (x \cdot x^{2})) + 25 x \cdot x^{3} + 25 x (- 10 x^{2}) + 25 x (25 x) - {(x)}^{2}$

Step 2.1.3.11.2

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

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Step 2.1.3.11.2.1

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

$V = x^{6} - 10 x^{5} + 25 x^{4} - 10 x^{5} + 100 x^{4} - 10 \cdot (25 (x \cdot x^{2})) + 25 x \cdot x^{3} + 25 x (- 10 x^{2}) + 25 x (25 x) - {(x)}^{2}$

Step 2.1.3.11.2.2

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

$V = x^{6} - 10 x^{5} + 25 x^{4} - 10 x^{5} + 100 x^{4} - 10 \cdot (25 x^{1 + 2}) + 25 x \cdot x^{3} + 25 x (- 10 x^{2}) + 25 x (25 x) - {(x)}^{2}$

Step 2.1.3.11.3

Add $1$ and $2$ .

$V = x^{6} - 10 x^{5} + 25 x^{4} - 10 x^{5} + 100 x^{4} - 10 \cdot (25 x^{3}) + 25 x \cdot x^{3} + 25 x (- 10 x^{2}) + 25 x (25 x) - {(x)}^{2}$

Step 2.1.3.12

Multiply $- 10$ by $25$ .

$V = x^{6} - 10 x^{5} + 25 x^{4} - 10 x^{5} + 100 x^{4} - 250 x^{3} + 25 x \cdot x^{3} + 25 x (- 10 x^{2}) + 25 x (25 x) - {(x)}^{2}$

Step 2.1.3.13

Multiply $x$ by $x^{3}$ by adding the exponents.

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Step 2.1.3.13.1

Move $x^{3}$ .

$V = x^{6} - 10 x^{5} + 25 x^{4} - 10 x^{5} + 100 x^{4} - 250 x^{3} + 25 (x^{3} x) + 25 x (- 10 x^{2}) + 25 x (25 x) - {(x)}^{2}$

Step 2.1.3.13.2

Multiply $x^{3}$ by $x$ .

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Step 2.1.3.13.2.1

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

$V = x^{6} - 10 x^{5} + 25 x^{4} - 10 x^{5} + 100 x^{4} - 250 x^{3} + 25 (x^{3} x) + 25 x (- 10 x^{2}) + 25 x (25 x) - {(x)}^{2}$

Step 2.1.3.13.2.2

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

$V = x^{6} - 10 x^{5} + 25 x^{4} - 10 x^{5} + 100 x^{4} - 250 x^{3} + 25 x^{3 + 1} + 25 x (- 10 x^{2}) + 25 x (25 x) - {(x)}^{2}$

Step 2.1.3.13.3

Add $3$ and $1$ .

$V = x^{6} - 10 x^{5} + 25 x^{4} - 10 x^{5} + 100 x^{4} - 250 x^{3} + 25 x^{4} + 25 x (- 10 x^{2}) + 25 x (25 x) - {(x)}^{2}$

Step 2.1.3.14

Rewrite using the commutative property of multiplication.

$V = x^{6} - 10 x^{5} + 25 x^{4} - 10 x^{5} + 100 x^{4} - 250 x^{3} + 25 x^{4} + 25 \cdot (- 10 x \cdot x^{2}) + 25 x (25 x) - {(x)}^{2}$

Step 2.1.3.15

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

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Step 2.1.3.15.1

Move $x^{2}$ .

$V = x^{6} - 10 x^{5} + 25 x^{4} - 10 x^{5} + 100 x^{4} - 250 x^{3} + 25 x^{4} + 25 \cdot (- 10 (x^{2} x)) + 25 x (25 x) - {(x)}^{2}$

Step 2.1.3.15.2

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

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Step 2.1.3.15.2.1

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

$V = x^{6} - 10 x^{5} + 25 x^{4} - 10 x^{5} + 100 x^{4} - 250 x^{3} + 25 x^{4} + 25 \cdot (- 10 (x^{2} x)) + 25 x (25 x) - {(x)}^{2}$

Step 2.1.3.15.2.2

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

$V = x^{6} - 10 x^{5} + 25 x^{4} - 10 x^{5} + 100 x^{4} - 250 x^{3} + 25 x^{4} + 25 \cdot (- 10 x^{2 + 1}) + 25 x (25 x) - {(x)}^{2}$

Step 2.1.3.15.3

Add $2$ and $1$ .

$V = x^{6} - 10 x^{5} + 25 x^{4} - 10 x^{5} + 100 x^{4} - 250 x^{3} + 25 x^{4} + 25 \cdot (- 10 x^{3}) + 25 x (25 x) - {(x)}^{2}$

Step 2.1.3.16

Multiply $25$ by $- 10$ .

$V = x^{6} - 10 x^{5} + 25 x^{4} - 10 x^{5} + 100 x^{4} - 250 x^{3} + 25 x^{4} - 250 x^{3} + 25 x (25 x) - {(x)}^{2}$

Step 2.1.3.17

Rewrite using the commutative property of multiplication.

$V = x^{6} - 10 x^{5} + 25 x^{4} - 10 x^{5} + 100 x^{4} - 250 x^{3} + 25 x^{4} - 250 x^{3} + 25 \cdot (25 x \cdot x) - {(x)}^{2}$

Step 2.1.3.18

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

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Step 2.1.3.18.1

Move $x$ .

$V = x^{6} - 10 x^{5} + 25 x^{4} - 10 x^{5} + 100 x^{4} - 250 x^{3} + 25 x^{4} - 250 x^{3} + 25 \cdot (25 (x \cdot x)) - {(x)}^{2}$

Step 2.1.3.18.2

Multiply $x$ by $x$ .

$V = x^{6} - 10 x^{5} + 25 x^{4} - 10 x^{5} + 100 x^{4} - 250 x^{3} + 25 x^{4} - 250 x^{3} + 25 \cdot (25 x^{2}) - {(x)}^{2}$

Step 2.1.3.19

Multiply $25$ by $25$ .

$V = x^{6} - 10 x^{5} + 25 x^{4} - 10 x^{5} + 100 x^{4} - 250 x^{3} + 25 x^{4} - 250 x^{3} + 625 x^{2} - {(x)}^{2}$

Step 2.1.4

Subtract $10 x^{5}$ from $- 10 x^{5}$ .

$V = x^{6} - 20 x^{5} + 25 x^{4} + 100 x^{4} - 250 x^{3} + 25 x^{4} - 250 x^{3} + 625 x^{2} - {(x)}^{2}$

Step 2.1.5

Add $25 x^{4}$ and $100 x^{4}$ .

$V = x^{6} - 20 x^{5} + 125 x^{4} - 250 x^{3} + 25 x^{4} - 250 x^{3} + 625 x^{2} - {(x)}^{2}$

Step 2.1.6

Add $125 x^{4}$ and $25 x^{4}$ .

$V = x^{6} - 20 x^{5} + 150 x^{4} - 250 x^{3} - 250 x^{3} + 625 x^{2} - {(x)}^{2}$

Step 2.1.7

Subtract $250 x^{3}$ from $- 250 x^{3}$ .

$V = x^{6} - 20 x^{5} + 150 x^{4} - 500 x^{3} + 625 x^{2} - x^{2}$

Step 2.2

Subtract $x^{2}$ from $625 x^{2}$ .

$V = x^{6} - 20 x^{5} + 150 x^{4} - 500 x^{3} + 624 x^{2}$

Step 3

Split the single integral into multiple integrals.

$V = π (\int_{0}^{4} x^{6} d x + \int_{0}^{4} - 20 x^{5} d x + \int_{0}^{4} 150 x^{4} d x + \int_{0}^{4} - 500 x^{3} d x + \int_{0}^{4} 624 x^{2} d x)$

Step 4

By the Power Rule, the integral of $x^{6}$ with respect to $x$ is $\frac{1}{7} x^{7}$ .

$V = π (\frac{1}{7} x^{7}]_{0}^{4} + \int_{0}^{4} - 20 x^{5} d x + \int_{0}^{4} 150 x^{4} d x + \int_{0}^{4} - 500 x^{3} d x + \int_{0}^{4} 624 x^{2} d x)$

Step 5

Combine $\frac{1}{7}$ and $x^{7}$ .

$V = π (\frac{x^{7}}{7}]_{0}^{4} + \int_{0}^{4} - 20 x^{5} d x + \int_{0}^{4} 150 x^{4} d x + \int_{0}^{4} - 500 x^{3} d x + \int_{0}^{4} 624 x^{2} d x)$

Step 6

Since $- 20$ is constant with respect to $x$ , move $- 20$ out of the integral.

$V = π (\frac{x^{7}}{7}]_{0}^{4} - 20 \int_{0}^{4} x^{5} d x + \int_{0}^{4} 150 x^{4} d x + \int_{0}^{4} - 500 x^{3} d x + \int_{0}^{4} 624 x^{2} d x)$

Step 7

By the Power Rule, the integral of $x^{5}$ with respect to $x$ is $\frac{1}{6} x^{6}$ .

$V = π (\frac{x^{7}}{7}]_{0}^{4} - 20 (\frac{1}{6} x^{6}]_{0}^{4}) + \int_{0}^{4} 150 x^{4} d x + \int_{0}^{4} - 500 x^{3} d x + \int_{0}^{4} 624 x^{2} d x)$

Step 8

Combine $\frac{1}{6}$ and $x^{6}$ .

$V = π (\frac{x^{7}}{7}]_{0}^{4} - 20 (\frac{x^{6}}{6}]_{0}^{4}) + \int_{0}^{4} 150 x^{4} d x + \int_{0}^{4} - 500 x^{3} d x + \int_{0}^{4} 624 x^{2} d x)$

Step 9

Since $150$ is constant with respect to $x$ , move $150$ out of the integral.

$V = π (\frac{x^{7}}{7}]_{0}^{4} - 20 (\frac{x^{6}}{6}]_{0}^{4}) + 150 \int_{0}^{4} x^{4} d x + \int_{0}^{4} - 500 x^{3} d x + \int_{0}^{4} 624 x^{2} d x)$

Step 10

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

$V = π (\frac{x^{7}}{7}]_{0}^{4} - 20 (\frac{x^{6}}{6}]_{0}^{4}) + 150 (\frac{1}{5} x^{5}]_{0}^{4}) + \int_{0}^{4} - 500 x^{3} d x + \int_{0}^{4} 624 x^{2} d x)$

Step 11

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

$V = π (\frac{x^{7}}{7}]_{0}^{4} - 20 (\frac{x^{6}}{6}]_{0}^{4}) + 150 (\frac{x^{5}}{5}]_{0}^{4}) + \int_{0}^{4} - 500 x^{3} d x + \int_{0}^{4} 624 x^{2} d x)$

Step 12

Since $- 500$ is constant with respect to $x$ , move $- 500$ out of the integral.

$V = π (\frac{x^{7}}{7}]_{0}^{4} - 20 (\frac{x^{6}}{6}]_{0}^{4}) + 150 (\frac{x^{5}}{5}]_{0}^{4}) - 500 \int_{0}^{4} x^{3} d x + \int_{0}^{4} 624 x^{2} d x)$

Step 13

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

$V = π (\frac{x^{7}}{7}]_{0}^{4} - 20 (\frac{x^{6}}{6}]_{0}^{4}) + 150 (\frac{x^{5}}{5}]_{0}^{4}) - 500 (\frac{1}{4} x^{4}]_{0}^{4}) + \int_{0}^{4} 624 x^{2} d x)$

Step 14

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

$V = π (\frac{x^{7}}{7}]_{0}^{4} - 20 (\frac{x^{6}}{6}]_{0}^{4}) + 150 (\frac{x^{5}}{5}]_{0}^{4}) - 500 (\frac{x^{4}}{4}]_{0}^{4}) + \int_{0}^{4} 624 x^{2} d x)$

Step 15

Since $624$ is constant with respect to $x$ , move $624$ out of the integral.

$V = π (\frac{x^{7}}{7}]_{0}^{4} - 20 (\frac{x^{6}}{6}]_{0}^{4}) + 150 (\frac{x^{5}}{5}]_{0}^{4}) - 500 (\frac{x^{4}}{4}]_{0}^{4}) + 624 \int_{0}^{4} x^{2} d x)$

Step 16

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

$V = π (\frac{x^{7}}{7}]_{0}^{4} - 20 (\frac{x^{6}}{6}]_{0}^{4}) + 150 (\frac{x^{5}}{5}]_{0}^{4}) - 500 (\frac{x^{4}}{4}]_{0}^{4}) + 624 (\frac{1}{3} x^{3}]_{0}^{4}))$

Step 17

Simplify the answer.

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Step 17.1

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

$V = π (\frac{x^{7}}{7}]_{0}^{4} - 20 (\frac{x^{6}}{6}]_{0}^{4}) + 150 (\frac{x^{5}}{5}]_{0}^{4}) - 500 (\frac{x^{4}}{4}]_{0}^{4}) + 624 (\frac{x^{3}}{3}]_{0}^{4}))$

Step 17.2

Substitute and simplify.

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Step 17.2.1

Evaluate $\frac{x^{7}}{7}$ at $4$ and at $0$ .

$V = π ((\frac{4^{7}}{7}) - \frac{0^{7}}{7} - 20 (\frac{x^{6}}{6}]_{0}^{4}) + 150 (\frac{x^{5}}{5}]_{0}^{4}) - 500 (\frac{x^{4}}{4}]_{0}^{4}) + 624 (\frac{x^{3}}{3}]_{0}^{4}))$

Step 17.2.2

Evaluate $\frac{x^{6}}{6}$ at $4$ and at $0$ .

$V = π (\frac{4^{7}}{7} - \frac{0^{7}}{7} - 20 (\frac{4^{6}}{6} - \frac{0^{6}}{6}) + 150 (\frac{x^{5}}{5}]_{0}^{4}) - 500 (\frac{x^{4}}{4}]_{0}^{4}) + 624 (\frac{x^{3}}{3}]_{0}^{4}))$

Step 17.2.3

Evaluate $\frac{x^{5}}{5}$ at $4$ and at $0$ .

$V = π (\frac{4^{7}}{7} - \frac{0^{7}}{7} - 20 (\frac{4^{6}}{6} - \frac{0^{6}}{6}) + 150 ((\frac{4^{5}}{5}) - \frac{0^{5}}{5}) - 500 (\frac{x^{4}}{4}]_{0}^{4}) + 624 (\frac{x^{3}}{3}]_{0}^{4}))$

Step 17.2.4

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

$V = π (\frac{4^{7}}{7} - \frac{0^{7}}{7} - 20 (\frac{4^{6}}{6} - \frac{0^{6}}{6}) + 150 (\frac{4^{5}}{5} - \frac{0^{5}}{5}) - 500 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 624 (\frac{x^{3}}{3}]_{0}^{4}))$

Step 17.2.5

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

Step 17.2.6

Simplify.

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Step 17.2.6.1

Raise $4$ to the power of $7$ .

$V = π (\frac{16384}{7} - \frac{0^{7}}{7} - 20 (\frac{4^{6}}{6} - \frac{0^{6}}{6}) + 150 (\frac{4^{5}}{5} - \frac{0^{5}}{5}) - 500 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.2

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

$V = π (\frac{16384}{7} - \frac{0}{7} - 20 (\frac{4^{6}}{6} - \frac{0^{6}}{6}) + 150 (\frac{4^{5}}{5} - \frac{0^{5}}{5}) - 500 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.3

Cancel the common factor of $0$ and $7$ .

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Step 17.2.6.3.1

Factor $7$ out of $0$ .

$V = π (\frac{16384}{7} - \frac{7 (0)}{7} - 20 (\frac{4^{6}}{6} - \frac{0^{6}}{6}) + 150 (\frac{4^{5}}{5} - \frac{0^{5}}{5}) - 500 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.3.2

Cancel the common factors.

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Step 17.2.6.3.2.1

Factor $7$ out of $7$ .

$V = π (\frac{16384}{7} - \frac{7 \cdot 0}{7 \cdot 1} - 20 (\frac{4^{6}}{6} - \frac{0^{6}}{6}) + 150 (\frac{4^{5}}{5} - \frac{0^{5}}{5}) - 500 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.3.2.2

Cancel the common factor.

Step 17.2.6.3.2.3

Rewrite the expression.

$V = π (\frac{16384}{7} - \frac{0}{1} - 20 (\frac{4^{6}}{6} - \frac{0^{6}}{6}) + 150 (\frac{4^{5}}{5} - \frac{0^{5}}{5}) - 500 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.3.2.4

Divide $0$ by $1$ .

$V = π (\frac{16384}{7} - 0 - 20 (\frac{4^{6}}{6} - \frac{0^{6}}{6}) + 150 (\frac{4^{5}}{5} - \frac{0^{5}}{5}) - 500 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.4

Multiply $- 1$ by $0$ .

$V = π (\frac{16384}{7} + 0 - 20 (\frac{4^{6}}{6} - \frac{0^{6}}{6}) + 150 (\frac{4^{5}}{5} - \frac{0^{5}}{5}) - 500 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.5

Add $\frac{16384}{7}$ and $0$ .

$V = π (\frac{16384}{7} - 20 (\frac{4^{6}}{6} - \frac{0^{6}}{6}) + 150 (\frac{4^{5}}{5} - \frac{0^{5}}{5}) - 500 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.6

Raise $4$ to the power of $6$ .

$V = π (\frac{16384}{7} - 20 (\frac{4096}{6} - \frac{0^{6}}{6}) + 150 (\frac{4^{5}}{5} - \frac{0^{5}}{5}) - 500 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.7

Cancel the common factor of $4096$ and $6$ .

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Step 17.2.6.7.1

Factor $2$ out of $4096$ .

$V = π (\frac{16384}{7} - 20 (\frac{2 (2048)}{6} - \frac{0^{6}}{6}) + 150 (\frac{4^{5}}{5} - \frac{0^{5}}{5}) - 500 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.7.2

Cancel the common factors.

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Step 17.2.6.7.2.1

Factor $2$ out of $6$ .

$V = π (\frac{16384}{7} - 20 (\frac{2 \cdot 2048}{2 \cdot 3} - \frac{0^{6}}{6}) + 150 (\frac{4^{5}}{5} - \frac{0^{5}}{5}) - 500 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.7.2.2

Cancel the common factor.

Step 17.2.6.7.2.3

Rewrite the expression.

$V = π (\frac{16384}{7} - 20 (\frac{2048}{3} - \frac{0^{6}}{6}) + 150 (\frac{4^{5}}{5} - \frac{0^{5}}{5}) - 500 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.8

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

$V = π (\frac{16384}{7} - 20 (\frac{2048}{3} - \frac{0}{6}) + 150 (\frac{4^{5}}{5} - \frac{0^{5}}{5}) - 500 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.9

Cancel the common factor of $0$ and $6$ .

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Step 17.2.6.9.1

Factor $6$ out of $0$ .

$V = π (\frac{16384}{7} - 20 (\frac{2048}{3} - \frac{6 (0)}{6}) + 150 (\frac{4^{5}}{5} - \frac{0^{5}}{5}) - 500 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.9.2

Cancel the common factors.

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Step 17.2.6.9.2.1

Factor $6$ out of $6$ .

$V = π (\frac{16384}{7} - 20 (\frac{2048}{3} - \frac{6 \cdot 0}{6 \cdot 1}) + 150 (\frac{4^{5}}{5} - \frac{0^{5}}{5}) - 500 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.9.2.2

Cancel the common factor.

Step 17.2.6.9.2.3

Rewrite the expression.

$V = π (\frac{16384}{7} - 20 (\frac{2048}{3} - \frac{0}{1}) + 150 (\frac{4^{5}}{5} - \frac{0^{5}}{5}) - 500 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.9.2.4

Divide $0$ by $1$ .

$V = π (\frac{16384}{7} - 20 (\frac{2048}{3} - 0) + 150 (\frac{4^{5}}{5} - \frac{0^{5}}{5}) - 500 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.10

Multiply $- 1$ by $0$ .

$V = π (\frac{16384}{7} - 20 (\frac{2048}{3} + 0) + 150 (\frac{4^{5}}{5} - \frac{0^{5}}{5}) - 500 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.11

Add $\frac{2048}{3}$ and $0$ .

$V = π (\frac{16384}{7} - 20 (\frac{2048}{3}) + 150 (\frac{4^{5}}{5} - \frac{0^{5}}{5}) - 500 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.12

Combine $- 20$ and $\frac{2048}{3}$ .

$V = π (\frac{16384}{7} + \frac{- 20 \cdot 2048}{3} + 150 (\frac{4^{5}}{5} - \frac{0^{5}}{5}) - 500 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.13

Multiply $- 20$ by $2048$ .

$V = π (\frac{16384}{7} + \frac{- 40960}{3} + 150 (\frac{4^{5}}{5} - \frac{0^{5}}{5}) - 500 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.14

Move the negative in front of the fraction.

$V = π (\frac{16384}{7} - \frac{40960}{3} + 150 (\frac{4^{5}}{5} - \frac{0^{5}}{5}) - 500 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.15

To write $\frac{16384}{7}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$V = π (\frac{16384}{7} \cdot \frac{3}{3} - \frac{40960}{3} + 150 (\frac{4^{5}}{5} - \frac{0^{5}}{5}) - 500 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.16

To write $- \frac{40960}{3}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$V = π (\frac{16384}{7} \cdot \frac{3}{3} - \frac{40960}{3} \cdot \frac{7}{7} + 150 (\frac{4^{5}}{5} - \frac{0^{5}}{5}) - 500 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.17

Write each expression with a common denominator of $21$ , by multiplying each by an appropriate factor of $1$ .

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Step 17.2.6.17.1

Multiply $\frac{16384}{7}$ by $\frac{3}{3}$ .

$V = π (\frac{16384 \cdot 3}{7 \cdot 3} - \frac{40960}{3} \cdot \frac{7}{7} + 150 (\frac{4^{5}}{5} - \frac{0^{5}}{5}) - 500 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.17.2

Multiply $7$ by $3$ .

$V = π (\frac{16384 \cdot 3}{21} - \frac{40960}{3} \cdot \frac{7}{7} + 150 (\frac{4^{5}}{5} - \frac{0^{5}}{5}) - 500 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.17.3

Multiply $\frac{40960}{3}$ by $\frac{7}{7}$ .

$V = π (\frac{16384 \cdot 3}{21} - \frac{40960 \cdot 7}{3 \cdot 7} + 150 (\frac{4^{5}}{5} - \frac{0^{5}}{5}) - 500 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.17.4

Multiply $3$ by $7$ .

$V = π (\frac{16384 \cdot 3}{21} - \frac{40960 \cdot 7}{21} + 150 (\frac{4^{5}}{5} - \frac{0^{5}}{5}) - 500 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.18

Combine the numerators over the common denominator.

$V = π (\frac{16384 \cdot 3 - 40960 \cdot 7}{21} + 150 (\frac{4^{5}}{5} - \frac{0^{5}}{5}) - 500 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.19

Simplify the numerator.

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Step 17.2.6.19.1

Multiply $16384$ by $3$ .

$V = π (\frac{49152 - 40960 \cdot 7}{21} + 150 (\frac{4^{5}}{5} - \frac{0^{5}}{5}) - 500 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.19.2

Multiply $- 40960$ by $7$ .

$V = π (\frac{49152 - 286720}{21} + 150 (\frac{4^{5}}{5} - \frac{0^{5}}{5}) - 500 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.19.3

Subtract $286720$ from $49152$ .

$V = π (\frac{- 237568}{21} + 150 (\frac{4^{5}}{5} - \frac{0^{5}}{5}) - 500 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.20

Move the negative in front of the fraction.

$V = π (- \frac{237568}{21} + 150 (\frac{4^{5}}{5} - \frac{0^{5}}{5}) - 500 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.21

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

$V = π (- \frac{237568}{21} + 150 (\frac{1024}{5} - \frac{0^{5}}{5}) - 500 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.22

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

$V = π (- \frac{237568}{21} + 150 (\frac{1024}{5} - \frac{0}{5}) - 500 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.23

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

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Step 17.2.6.23.1

Factor $5$ out of $0$ .

$V = π (- \frac{237568}{21} + 150 (\frac{1024}{5} - \frac{5 (0)}{5}) - 500 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.23.2

Cancel the common factors.

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Step 17.2.6.23.2.1

Factor $5$ out of $5$ .

$V = π (- \frac{237568}{21} + 150 (\frac{1024}{5} - \frac{5 \cdot 0}{5 \cdot 1}) - 500 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.23.2.2

Cancel the common factor.

$V = π (- \frac{237568}{21} + 150 (\frac{1024}{5} - \frac{5 \cdot 0}{5 \cdot 1}) - 500 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.23.2.3

Rewrite the expression.

$V = π (- \frac{237568}{21} + 150 (\frac{1024}{5} - \frac{0}{1}) - 500 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.23.2.4

Divide $0$ by $1$ .

$V = π (- \frac{237568}{21} + 150 (\frac{1024}{5} - 0) - 500 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.24

Multiply $- 1$ by $0$ .

$V = π (- \frac{237568}{21} + 150 (\frac{1024}{5} + 0) - 500 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.25

Add $\frac{1024}{5}$ and $0$ .

$V = π (- \frac{237568}{21} + 150 (\frac{1024}{5}) - 500 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.26

Combine $150$ and $\frac{1024}{5}$ .

$V = π (- \frac{237568}{21} + \frac{150 \cdot 1024}{5} - 500 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.27

Multiply $150$ by $1024$ .

$V = π (- \frac{237568}{21} + \frac{153600}{5} - 500 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.28

Cancel the common factor of $153600$ and $5$ .

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Step 17.2.6.28.1

Factor $5$ out of $153600$ .

$V = π (- \frac{237568}{21} + \frac{5 \cdot 30720}{5} - 500 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.28.2

Cancel the common factors.

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Step 17.2.6.28.2.1

Factor $5$ out of $5$ .

$V = π (- \frac{237568}{21} + \frac{5 \cdot 30720}{5 (1)} - 500 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.28.2.2

Cancel the common factor.

$V = π (- \frac{237568}{21} + \frac{5 \cdot 30720}{5 \cdot 1} - 500 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.28.2.3

Rewrite the expression.

$V = π (- \frac{237568}{21} + \frac{30720}{1} - 500 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.28.2.4

Divide $30720$ by $1$ .

$V = π (- \frac{237568}{21} + 30720 - 500 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.29

To write $30720$ as a fraction with a common denominator, multiply by $\frac{21}{21}$ .

$V = π (- \frac{237568}{21} + 30720 \cdot \frac{21}{21} - 500 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.30

Combine $30720$ and $\frac{21}{21}$ .

$V = π (- \frac{237568}{21} + \frac{30720 \cdot 21}{21} - 500 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.31

Combine the numerators over the common denominator.

$V = π (\frac{- 237568 + 30720 \cdot 21}{21} - 500 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.32

Simplify the numerator.

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Step 17.2.6.32.1

Multiply $30720$ by $21$ .

$V = π (\frac{- 237568 + 645120}{21} - 500 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.32.2

Add $- 237568$ and $645120$ .

$V = π (\frac{407552}{21} - 500 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.33

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

$V = π (\frac{407552}{21} - 500 (\frac{256}{4} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.34

Cancel the common factor of $256$ and $4$ .

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Step 17.2.6.34.1

Factor $4$ out of $256$ .

$V = π (\frac{407552}{21} - 500 (\frac{4 \cdot 64}{4} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.34.2

Cancel the common factors.

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Step 17.2.6.34.2.1

Factor $4$ out of $4$ .

$V = π (\frac{407552}{21} - 500 (\frac{4 \cdot 64}{4 (1)} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.34.2.2

Cancel the common factor.

$V = π (\frac{407552}{21} - 500 (\frac{4 \cdot 64}{4 \cdot 1} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.34.2.3

Rewrite the expression.

$V = π (\frac{407552}{21} - 500 (\frac{64}{1} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.34.2.4

Divide $64$ by $1$ .

$V = π (\frac{407552}{21} - 500 (64 - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.35

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

$V = π (\frac{407552}{21} - 500 (64 - \frac{0}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.36

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

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Step 17.2.6.36.1

Factor $4$ out of $0$ .

$V = π (\frac{407552}{21} - 500 (64 - \frac{4 (0)}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.36.2

Cancel the common factors.

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Step 17.2.6.36.2.1

Factor $4$ out of $4$ .

$V = π (\frac{407552}{21} - 500 (64 - \frac{4 \cdot 0}{4 \cdot 1}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.36.2.2

Cancel the common factor.

$V = π (\frac{407552}{21} - 500 (64 - \frac{4 \cdot 0}{4 \cdot 1}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.36.2.3

Rewrite the expression.

$V = π (\frac{407552}{21} - 500 (64 - \frac{0}{1}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.36.2.4

Divide $0$ by $1$ .

$V = π (\frac{407552}{21} - 500 (64 - 0) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.37

Multiply $- 1$ by $0$ .

$V = π (\frac{407552}{21} - 500 (64 + 0) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.38

Add $64$ and $0$ .

$V = π (\frac{407552}{21} - 500 \cdot 64 + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.39

Multiply $- 500$ by $64$ .

$V = π (\frac{407552}{21} - 32000 + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.40

To write $- 32000$ as a fraction with a common denominator, multiply by $\frac{21}{21}$ .

$V = π (\frac{407552}{21} - 32000 \cdot \frac{21}{21} + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.41

Combine $- 32000$ and $\frac{21}{21}$ .

$V = π (\frac{407552}{21} + \frac{- 32000 \cdot 21}{21} + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.42

Combine the numerators over the common denominator.

$V = π (\frac{407552 - 32000 \cdot 21}{21} + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.43

Simplify the numerator.

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Step 17.2.6.43.1

Multiply $- 32000$ by $21$ .

$V = π (\frac{407552 - 672000}{21} + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.43.2

Subtract $672000$ from $407552$ .

$V = π (\frac{- 264448}{21} + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.44

Move the negative in front of the fraction.

$V = π (- \frac{264448}{21} + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Step 17.2.6.45

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

$V = π (- \frac{264448}{21} + 624 (\frac{64}{3} - \frac{0^{3}}{3}))$

Step 17.2.6.46

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

$V = π (- \frac{264448}{21} + 624 (\frac{64}{3} - \frac{0}{3}))$

Step 17.2.6.47

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

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Step 17.2.6.47.1

Factor $3$ out of $0$ .

$V = π (- \frac{264448}{21} + 624 (\frac{64}{3} - \frac{3 (0)}{3}))$

Step 17.2.6.47.2

Cancel the common factors.

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Step 17.2.6.47.2.1

Factor $3$ out of $3$ .

$V = π (- \frac{264448}{21} + 624 (\frac{64}{3} - \frac{3 \cdot 0}{3 \cdot 1}))$

Step 17.2.6.47.2.2

Cancel the common factor.

$V = π (- \frac{264448}{21} + 624 (\frac{64}{3} - \frac{3 \cdot 0}{3 \cdot 1}))$

Step 17.2.6.47.2.3

Rewrite the expression.

$V = π (- \frac{264448}{21} + 624 (\frac{64}{3} - \frac{0}{1}))$

Step 17.2.6.47.2.4

Divide $0$ by $1$ .

$V = π (- \frac{264448}{21} + 624 (\frac{64}{3} - 0))$

Step 17.2.6.48

Multiply $- 1$ by $0$ .

$V = π (- \frac{264448}{21} + 624 (\frac{64}{3} + 0))$

Step 17.2.6.49

Add $\frac{64}{3}$ and $0$ .

$V = π (- \frac{264448}{21} + 624 (\frac{64}{3}))$

Step 17.2.6.50

Combine $624$ and $\frac{64}{3}$ .

$V = π (- \frac{264448}{21} + \frac{624 \cdot 64}{3})$

Step 17.2.6.51

Multiply $624$ by $64$ .

$V = π (- \frac{264448}{21} + \frac{39936}{3})$

Step 17.2.6.52

Cancel the common factor of $39936$ and $3$ .

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Step 17.2.6.52.1

Factor $3$ out of $39936$ .

$V = π (- \frac{264448}{21} + \frac{3 \cdot 13312}{3})$

Step 17.2.6.52.2

Cancel the common factors.

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Step 17.2.6.52.2.1

Factor $3$ out of $3$ .

$V = π (- \frac{264448}{21} + \frac{3 \cdot 13312}{3 (1)})$

Step 17.2.6.52.2.2

Cancel the common factor.

$V = π (- \frac{264448}{21} + \frac{3 \cdot 13312}{3 \cdot 1})$

Step 17.2.6.52.2.3

Rewrite the expression.

$V = π (- \frac{264448}{21} + \frac{13312}{1})$

Step 17.2.6.52.2.4

Divide $13312$ by $1$ .

$V = π (- \frac{264448}{21} + 13312)$

Step 17.2.6.53

To write $13312$ as a fraction with a common denominator, multiply by $\frac{21}{21}$ .

$V = π (- \frac{264448}{21} + 13312 \cdot \frac{21}{21})$

Step 17.2.6.54

Combine $13312$ and $\frac{21}{21}$ .

$V = π (- \frac{264448}{21} + \frac{13312 \cdot 21}{21})$

Step 17.2.6.55

Combine the numerators over the common denominator.

$V = π (\frac{- 264448 + 13312 \cdot 21}{21})$

Step 17.2.6.56

Simplify the numerator.

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Step 17.2.6.56.1

Multiply $13312$ by $21$ .

$V = π (\frac{- 264448 + 279552}{21})$

Step 17.2.6.56.2

Add $- 264448$ and $279552$ .

$V = π (\frac{15104}{21})$

Step 17.2.6.57

Combine $π$ and $\frac{15104}{21}$ .

$V = \frac{π \cdot 15104}{21}$

Step 17.2.6.58

Move $15104$ to the left of $π$ .

$V = \frac{15104 π}{21}$

Step 18

The result can be shown in multiple forms.

Exact Form:

$V = \frac{15104 π}{21}$

Decimal Form:

$V = 2259.55311618 \dots$

Step 19

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