Calculus Examples

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

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 $A = π r^{2}$ .

$V = π \int_{- 1}^{2} {(f (x))}^{2} - {(g (x))}^{2} d x$ where $f (x) = x + 2$ and $g (x) = x^{2}$

Step 2

Simplify each term.

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Step 2.1

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

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

Step 2.2

Expand $(x + 2) (x + 2)$ using the FOIL Method.

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Step 2.2.1

Apply the distributive property.

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

Step 2.2.2

Apply the distributive property.

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

Step 2.2.3

Apply the distributive property.

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

Step 2.3

Simplify and combine like terms.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Multiply $x$ by $x$ .

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

Step 2.3.1.2

Move $2$ to the left of $x$ .

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

Step 2.3.1.3

Multiply $2$ by $2$ .

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

Step 2.3.2

Add $2 x$ and $2 x$ .

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

Step 2.4

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 2.4.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.4.2

Multiply $2$ by $2$ .

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

Step 3

Split the single integral into multiple integrals.

$V = π (\int_{- 1}^{2} x^{2} d x + \int_{- 1}^{2} 4 x d x + \int_{- 1}^{2} 4 d x + \int_{- 1}^{2} - x^{4} d x)$

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Apply the constant rule.

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

Step 10

Combine $\frac{x^{3}}{3}]_{- 1}^{2}$ and $4 x]_{- 1}^{2}$ .

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

Step 11

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

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

Step 12

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

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

Step 13

Simplify the answer.

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Step 13.1

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

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

Step 13.2

Substitute and simplify.

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Step 13.2.1

Evaluate $\frac{x^{2}}{2}$ at $2$ and at $- 1$ .

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

Step 13.2.2

Evaluate $\frac{x^{3}}{3} + 4 x$ at $2$ and at $- 1$ .

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

Step 13.2.3

Evaluate $\frac{x^{5}}{5}$ at $2$ and at $- 1$ .

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

Step 13.2.4

Simplify.

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Step 13.2.4.1

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

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

Step 13.2.4.2

Cancel the common factor of $4$ and $2$ .

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Step 13.2.4.2.1

Factor $2$ out of $4$ .

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

Step 13.2.4.2.2

Cancel the common factors.

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Step 13.2.4.2.2.1

Factor $2$ out of $2$ .

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

Step 13.2.4.2.2.2

Cancel the common factor.

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

Step 13.2.4.2.2.3

Rewrite the expression.

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

Step 13.2.4.2.2.4

Divide $2$ by $1$ .

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

Step 13.2.4.3

Raise $- 1$ to the power of $2$ .

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

Step 13.2.4.4

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 13.2.4.5

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

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

Step 13.2.4.6

Combine the numerators over the common denominator.

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

Step 13.2.4.7

Simplify the numerator.

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Step 13.2.4.7.1

Multiply $2$ by $2$ .

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

Step 13.2.4.7.2

Subtract $1$ from $4$ .

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

Step 13.2.4.8

Combine $4$ and $\frac{3}{2}$ .

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

Step 13.2.4.9

Multiply $4$ by $3$ .

$V = π (\frac{12}{2} + \frac{2^{3}}{3} + 4 \cdot 2 - (\frac{{(- 1)}^{3}}{3} + 4 \cdot - 1) - (\frac{2^{5}}{5} - \frac{{(- 1)}^{5}}{5}))$

Step 13.2.4.10

Cancel the common factor of $12$ and $2$ .

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Step 13.2.4.10.1

Factor $2$ out of $12$ .

$V = π (\frac{2 \cdot 6}{2} + \frac{2^{3}}{3} + 4 \cdot 2 - (\frac{{(- 1)}^{3}}{3} + 4 \cdot - 1) - (\frac{2^{5}}{5} - \frac{{(- 1)}^{5}}{5}))$

Step 13.2.4.10.2

Cancel the common factors.

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Step 13.2.4.10.2.1

Factor $2$ out of $2$ .

$V = π (\frac{2 \cdot 6}{2 (1)} + \frac{2^{3}}{3} + 4 \cdot 2 - (\frac{{(- 1)}^{3}}{3} + 4 \cdot - 1) - (\frac{2^{5}}{5} - \frac{{(- 1)}^{5}}{5}))$

Step 13.2.4.10.2.2

Cancel the common factor.

$V = π (\frac{2 \cdot 6}{2 \cdot 1} + \frac{2^{3}}{3} + 4 \cdot 2 - (\frac{{(- 1)}^{3}}{3} + 4 \cdot - 1) - (\frac{2^{5}}{5} - \frac{{(- 1)}^{5}}{5}))$

Step 13.2.4.10.2.3

Rewrite the expression.

$V = π (\frac{6}{1} + \frac{2^{3}}{3} + 4 \cdot 2 - (\frac{{(- 1)}^{3}}{3} + 4 \cdot - 1) - (\frac{2^{5}}{5} - \frac{{(- 1)}^{5}}{5}))$

Step 13.2.4.10.2.4

Divide $6$ by $1$ .

$V = π (6 + \frac{2^{3}}{3} + 4 \cdot 2 - (\frac{{(- 1)}^{3}}{3} + 4 \cdot - 1) - (\frac{2^{5}}{5} - \frac{{(- 1)}^{5}}{5}))$

Step 13.2.4.11

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

$V = π (6 + \frac{8}{3} + 4 \cdot 2 - (\frac{{(- 1)}^{3}}{3} + 4 \cdot - 1) - (\frac{2^{5}}{5} - \frac{{(- 1)}^{5}}{5}))$

Step 13.2.4.12

Multiply $4$ by $2$ .

$V = π (6 + \frac{8}{3} + 8 - (\frac{{(- 1)}^{3}}{3} + 4 \cdot - 1) - (\frac{2^{5}}{5} - \frac{{(- 1)}^{5}}{5}))$

Step 13.2.4.13

To write $8$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$V = π (6 + \frac{8}{3} + 8 \cdot \frac{3}{3} - (\frac{{(- 1)}^{3}}{3} + 4 \cdot - 1) - (\frac{2^{5}}{5} - \frac{{(- 1)}^{5}}{5}))$

Step 13.2.4.14

Combine $8$ and $\frac{3}{3}$ .

$V = π (6 + \frac{8}{3} + \frac{8 \cdot 3}{3} - (\frac{{(- 1)}^{3}}{3} + 4 \cdot - 1) - (\frac{2^{5}}{5} - \frac{{(- 1)}^{5}}{5}))$

Step 13.2.4.15

Combine the numerators over the common denominator.

$V = π (6 + \frac{8 + 8 \cdot 3}{3} - (\frac{{(- 1)}^{3}}{3} + 4 \cdot - 1) - (\frac{2^{5}}{5} - \frac{{(- 1)}^{5}}{5}))$

Step 13.2.4.16

Simplify the numerator.

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Step 13.2.4.16.1

Multiply $8$ by $3$ .

$V = π (6 + \frac{8 + 24}{3} - (\frac{{(- 1)}^{3}}{3} + 4 \cdot - 1) - (\frac{2^{5}}{5} - \frac{{(- 1)}^{5}}{5}))$

Step 13.2.4.16.2

Add $8$ and $24$ .

$V = π (6 + \frac{32}{3} - (\frac{{(- 1)}^{3}}{3} + 4 \cdot - 1) - (\frac{2^{5}}{5} - \frac{{(- 1)}^{5}}{5}))$

Step 13.2.4.17

Raise $- 1$ to the power of $3$ .

$V = π (6 + \frac{32}{3} - (\frac{- 1}{3} + 4 \cdot - 1) - (\frac{2^{5}}{5} - \frac{{(- 1)}^{5}}{5}))$

Step 13.2.4.18

Move the negative in front of the fraction.

$V = π (6 + \frac{32}{3} - (- \frac{1}{3} + 4 \cdot - 1) - (\frac{2^{5}}{5} - \frac{{(- 1)}^{5}}{5}))$

Step 13.2.4.19

Multiply $4$ by $- 1$ .

$V = π (6 + \frac{32}{3} - (- \frac{1}{3} - 4) - (\frac{2^{5}}{5} - \frac{{(- 1)}^{5}}{5}))$

Step 13.2.4.20

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$V = π (6 + \frac{32}{3} - (- \frac{1}{3} - 4 \cdot \frac{3}{3}) - (\frac{2^{5}}{5} - \frac{{(- 1)}^{5}}{5}))$

Step 13.2.4.21

Combine $- 4$ and $\frac{3}{3}$ .

$V = π (6 + \frac{32}{3} - (- \frac{1}{3} + \frac{- 4 \cdot 3}{3}) - (\frac{2^{5}}{5} - \frac{{(- 1)}^{5}}{5}))$

Step 13.2.4.22

Combine the numerators over the common denominator.

$V = π (6 + \frac{32}{3} - \frac{- 1 - 4 \cdot 3}{3} - (\frac{2^{5}}{5} - \frac{{(- 1)}^{5}}{5}))$

Step 13.2.4.23

Simplify the numerator.

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Step 13.2.4.23.1

Multiply $- 4$ by $3$ .

$V = π (6 + \frac{32}{3} - \frac{- 1 - 12}{3} - (\frac{2^{5}}{5} - \frac{{(- 1)}^{5}}{5}))$

Step 13.2.4.23.2

Subtract $12$ from $- 1$ .

$V = π (6 + \frac{32}{3} - \frac{- 13}{3} - (\frac{2^{5}}{5} - \frac{{(- 1)}^{5}}{5}))$

Step 13.2.4.24

Move the negative in front of the fraction.

$V = π (6 + \frac{32}{3} + \frac{13}{3} - (\frac{2^{5}}{5} - \frac{{(- 1)}^{5}}{5}))$

Step 13.2.4.25

Multiply $- 1$ by $- 1$ .

$V = π (6 + \frac{32}{3} + 1 (\frac{13}{3}) - (\frac{2^{5}}{5} - \frac{{(- 1)}^{5}}{5}))$

Step 13.2.4.26

Multiply $\frac{13}{3}$ by $1$ .

$V = π (6 + \frac{32}{3} + \frac{13}{3} - (\frac{2^{5}}{5} - \frac{{(- 1)}^{5}}{5}))$

Step 13.2.4.27

Combine the numerators over the common denominator.

$V = π (6 + \frac{32 + 13}{3} - (\frac{2^{5}}{5} - \frac{{(- 1)}^{5}}{5}))$

Step 13.2.4.28

Add $32$ and $13$ .

$V = π (6 + \frac{45}{3} - (\frac{2^{5}}{5} - \frac{{(- 1)}^{5}}{5}))$

Step 13.2.4.29

Cancel the common factor of $45$ and $3$ .

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Step 13.2.4.29.1

Factor $3$ out of $45$ .

$V = π (6 + \frac{3 \cdot 15}{3} - (\frac{2^{5}}{5} - \frac{{(- 1)}^{5}}{5}))$

Step 13.2.4.29.2

Cancel the common factors.

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Step 13.2.4.29.2.1

Factor $3$ out of $3$ .

$V = π (6 + \frac{3 \cdot 15}{3 (1)} - (\frac{2^{5}}{5} - \frac{{(- 1)}^{5}}{5}))$

Step 13.2.4.29.2.2

Cancel the common factor.

$V = π (6 + \frac{3 \cdot 15}{3 \cdot 1} - (\frac{2^{5}}{5} - \frac{{(- 1)}^{5}}{5}))$

Step 13.2.4.29.2.3

Rewrite the expression.

$V = π (6 + \frac{15}{1} - (\frac{2^{5}}{5} - \frac{{(- 1)}^{5}}{5}))$

Step 13.2.4.29.2.4

Divide $15$ by $1$ .

$V = π (6 + 15 - (\frac{2^{5}}{5} - \frac{{(- 1)}^{5}}{5}))$

Step 13.2.4.30

Add $6$ and $15$ .

$V = π (21 - (\frac{2^{5}}{5} - \frac{{(- 1)}^{5}}{5}))$

Step 13.2.4.31

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

$V = π (21 - (\frac{32}{5} - \frac{{(- 1)}^{5}}{5}))$

Step 13.2.4.32

Raise $- 1$ to the power of $5$ .

$V = π (21 - (\frac{32}{5} - \frac{- 1}{5}))$

Step 13.2.4.33

Move the negative in front of the fraction.

$V = π (21 - (\frac{32}{5} + \frac{1}{5}))$

Step 13.2.4.34

Multiply $- 1$ by $- 1$ .

$V = π (21 - (\frac{32}{5} + 1 (\frac{1}{5})))$

Step 13.2.4.35

Multiply $\frac{1}{5}$ by $1$ .

$V = π (21 - (\frac{32}{5} + \frac{1}{5}))$

Step 13.2.4.36

Combine the numerators over the common denominator.

$V = π (21 - \frac{32 + 1}{5})$

Step 13.2.4.37

Add $32$ and $1$ .

$V = π (21 - \frac{33}{5})$

Step 13.2.4.38

To write $21$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$V = π (21 \cdot \frac{5}{5} - \frac{33}{5})$

Step 13.2.4.39

Combine $21$ and $\frac{5}{5}$ .

$V = π (\frac{21 \cdot 5}{5} - \frac{33}{5})$

Step 13.2.4.40

Combine the numerators over the common denominator.

$V = π (\frac{21 \cdot 5 - 33}{5})$

Step 13.2.4.41

Simplify the numerator.

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Step 13.2.4.41.1

Multiply $21$ by $5$ .

$V = π (\frac{105 - 33}{5})$

Step 13.2.4.41.2

Subtract $33$ from $105$ .

$V = π (\frac{72}{5})$

Step 13.2.4.42

Combine $π$ and $\frac{72}{5}$ .

$V = \frac{π \cdot 72}{5}$

Step 13.2.4.43

Move $72$ to the left of $π$ .

$V = \frac{72 π}{5}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$V = \frac{72 π}{5}$

Decimal Form:

$V = 45.23893421 \dots$

Step 15