Calculus Examples

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

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 $g (x) = x$

Simplify the integrand.

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Simplify each term.

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

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

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.

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

Remove unnecessary parentheses.

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

Simplify each term.

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

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

Add $3$ and $3$ to get $6$ .

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

Move $x^{3}$ .

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

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

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

Add $3$ and $2$ to get $5$ .

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

Move $x^{3}$ .

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

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

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

Add $3$ and $1$ to get $4$ .

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

Move $x^{3}$ .

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

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

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

Add $3$ and $2$ to get $5$ .

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

Move $x^{2}$ .

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

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

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

Add $2$ and $2$ to get $4$ .

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

Multiply $- 10$ by $- 10$ to get $100$ .

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

Move $x^{2}$ .

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

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

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

Add $2$ and $1$ to get $3$ .

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

Multiply $25$ by $- 10$ to get $- 250$ .

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

Move $x^{3}$ .

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

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

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

Add $3$ and $1$ to get $4$ .

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

Move $x$ .

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

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

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

Add $1$ and $2$ to get $3$ .

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

Multiply $- 10$ by $25$ to get $- 250$ .

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

Move $x$ .

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

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

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

Add $1$ and $1$ to get $2$ .

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

Multiply $25$ by $25$ to get $625$ .

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

Subtract $10 x^{5}$ from $- 10 x^{5}$ to get $- 20 x^{5}$ .

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

Add $25 x^{4}$ and $100 x^{4}$ to get $125 x^{4}$ .

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

Add $125 x^{4}$ and $25 x^{4}$ to get $150 x^{4}$ .

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

Subtract $250 x^{3}$ from $- 250 x^{3}$ to get $- 500 x^{3}$ .

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

Remove parentheses around $x$ .

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

Subtract $x^{2}$ from $625 x^{2}$ to get $624 x^{2}$ .

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

Since integration is linear, the integral of $x^{6} - 20 x^{5} + 150 x^{4} - 500 x^{3} + 624 x^{2}$ with respect to $x$ is $\int_{0}^{4} x^{6} d x + \int_{0}^{4} - 20 x^{5} d x + \int_{0}^{4} 150 x^{4} d x + \int_{0}^{4} - 500 x^{3} d x + \int_{0}^{4} 624 x^{2} d x$ .

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

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

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

Combine fractions.

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Write $x^{7}$ as a fraction with denominator $1$ .

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

Multiply $\frac{1}{7}$ and $\frac{x^{7}}{1}$ to get $\frac{x^{7}}{7}$ .

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

Since $- 20$ is constant with respect to $x$ , the integral of $- 20 x^{5}$ with respect to $x$ is $- 20 \int_{0}^{4} x^{5} d x$ .

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

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

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

Combine fractions.

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Write $x^{6}$ as a fraction with denominator $1$ .

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

Multiply $\frac{1}{6}$ and $\frac{x^{6}}{1}$ to get $\frac{x^{6}}{6}$ .

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

Since $150$ is constant with respect to $x$ , the integral of $150 x^{4}$ with respect to $x$ is $150 \int_{0}^{4} x^{4} d x$ .

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

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

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

Combine fractions.

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Write $x^{5}$ as a fraction with denominator $1$ .

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

Multiply $\frac{1}{5}$ and $\frac{x^{5}}{1}$ to get $\frac{x^{5}}{5}$ .

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

Since $- 500$ is constant with respect to $x$ , the integral of $- 500 x^{3}$ with respect to $x$ is $- 500 \int_{0}^{4} x^{3} d x$ .

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

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

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

Combine fractions.

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Write $x^{4}$ as a fraction with denominator $1$ .

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

Multiply $\frac{1}{4}$ and $\frac{x^{4}}{1}$ to get $\frac{x^{4}}{4}$ .

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

Since $624$ is constant with respect to $x$ , the integral of $624 x^{2}$ with respect to $x$ is $624 \int_{0}^{4} x^{2} d x$ .

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

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

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

Simplify the answer.

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Write $x^{3}$ as a fraction with denominator $1$ .

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

Multiply $\frac{1}{3}$ and $\frac{x^{3}}{1}$ to get $\frac{x^{3}}{3}$ .

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

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

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

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

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

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

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

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

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

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

Raise $4$ to the power of $7$ to get $16384$ .

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

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

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

Reduce the expression by cancelling the common factors.

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Factor $7$ out of $7$ .

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

Cancel the common factor.

Rewrite the expression.

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

Divide $0$ by $1$ to get $0$ .

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

Multiply $- 1$ by $0$ to get $0$ .

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

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

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

Raise $4$ to the power of $6$ to get $4096$ .

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

Reduce the expression by cancelling the common factors.

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Factor $2$ out of $6$ .

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

Cancel the common factor.

Rewrite the expression.

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

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

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

Reduce the expression by cancelling the common factors.

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Factor $6$ out of $6$ .

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

Cancel the common factor.

Rewrite the expression.

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

Divide $0$ by $1$ to get $0$ .

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

Multiply $- 1$ by $0$ to get $0$ .

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

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

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

Write $- 20$ as a fraction with denominator $1$ .

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

Multiply $\frac{- 20}{1}$ and $\frac{2048}{3}$ to get $\frac{- 20 \cdot 2048}{3}$ .

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

Multiply $- 20$ by $2048$ to get $- 40960$ .

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

Move the negative in front of the fraction.

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

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

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

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

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

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

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Combine.

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

Multiply $7$ by $3$ to get $21$ .

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

Combine.

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

Multiply $3$ by $7$ to get $21$ .

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

Combine the numerators over the common denominator.

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

Multiply $16384$ by $3$ to get $49152$ .

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

Multiply $40960$ by $7$ to get $286720$ .

$V = π (\frac{49152 - 1 \cdot 286720}{21} + 150 (\frac{4^{5}}{5} - \frac{0^{5}}{5}) - 500 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Multiply $- 1$ by $286720$ to get $- 286720$ .

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

Subtract $286720$ from $49152$ to get $- 237568$ .

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

Move the negative in front of the fraction.

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

Raise $4$ to the power of $5$ to get $1024$ .

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

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

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

Reduce the expression by cancelling the common factors.

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Factor $5$ out of $5$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $0$ by $1$ to get $0$ .

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

Multiply $- 1$ by $0$ to get $0$ .

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

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

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

Write $150$ as a fraction with denominator $1$ .

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

Multiply $\frac{150}{1}$ and $\frac{1024}{5}$ to get $\frac{150 \cdot 1024}{5}$ .

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

Multiply $150$ by $1024$ to get $153600$ .

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

Reduce the expression by cancelling the common factors.

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Factor $5$ out of $5$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $30720$ by $1$ to get $30720$ .

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

To write $\frac{30720}{1}$ as a fraction with a common denominator, multiply by $\frac{21}{21}$ .

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

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

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Combine.

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

Multiply $21$ by $1$ to get $21$ .

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

Combine the numerators over the common denominator.

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

Multiply $- 1$ by $237568$ to get $- 237568$ .

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

Multiply $30720$ by $21$ to get $645120$ .

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

Add $- 237568$ and $645120$ to get $407552$ .

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

Raise $4$ to the power of $4$ to get $256$ .

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

Reduce the expression by cancelling the common factors.

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Factor $4$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $64$ by $1$ to get $64$ .

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

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

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

Reduce the expression by cancelling the common factors.

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Factor $4$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $0$ by $1$ to get $0$ .

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

Multiply $- 1$ by $0$ to get $0$ .

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

Add $64$ and $0$ to get $64$ .

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

Multiply $- 500$ by $64$ to get $- 32000$ .

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

To write $\frac{- 32000}{1}$ as a fraction with a common denominator, multiply by $\frac{21}{21}$ .

$V = π (\frac{407552}{21} + \frac{- 32000}{1} \cdot \frac{21}{21} + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

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

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Combine.

$V = π (\frac{407552}{21} + \frac{- 32000 \cdot 21}{1 \cdot 21} + 624 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}))$

Multiply $21$ by $1$ to get $21$ .

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

Combine the numerators over the common denominator.

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

Multiply $- 32000$ by $21$ to get $- 672000$ .

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

Subtract $672000$ from $407552$ to get $- 264448$ .

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

Move the negative in front of the fraction.

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

Raise $4$ to the power of $3$ to get $64$ .

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

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

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

Reduce the expression by cancelling the common factors.

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Factor $3$ out of $3$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $0$ by $1$ to get $0$ .

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

Multiply $- 1$ by $0$ to get $0$ .

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

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

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

Write $624$ as a fraction with denominator $1$ .

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

Multiply $\frac{624}{1}$ and $\frac{64}{3}$ to get $\frac{624 \cdot 64}{3}$ .

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

Multiply $624$ by $64$ to get $39936$ .

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

Reduce the expression by cancelling the common factors.

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Factor $3$ out of $3$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $13312$ by $1$ to get $13312$ .

$V = π (- \frac{264448}{21} + 13312)$

To write $\frac{13312}{1}$ as a fraction with a common denominator, multiply by $\frac{21}{21}$ .

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

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

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Combine.

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

Multiply $21$ by $1$ to get $21$ .

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

Combine the numerators over the common denominator.

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

Multiply $- 1$ by $264448$ to get $- 264448$ .

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

Multiply $13312$ by $21$ to get $279552$ .

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

Add $- 264448$ and $279552$ to get $15104$ .

$V = π (\frac{15104}{21})$

Write $π$ as a fraction with denominator $1$ .

$V = \frac{π}{1} \cdot \frac{15104}{21}$

Multiply $\frac{π}{1}$ and $\frac{15104}{21}$ to get $\frac{π \cdot 15104}{21}$ .

$V = \frac{π \cdot 15104}{21}$

Move $15104$ to the left of the expression $π \cdot 15104$ .

$V = \frac{15104 \cdot π}{21}$

Reorder terms.

$V = \frac{15104 π}{21}$

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