Calculus Examples

$\int_{0}^{4} x^{2} \sqrt{16 - x^{2}} d x$

Step 1

Let $x = 4 \sin (t)$ , where $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then $d x = 4 \cos (t) d t$ . Note that since $- \frac{π}{2} \leq t \leq \frac{π}{2}$ , $4 \cos (t)$ is positive.

$\int_{0}^{\frac{π}{2}} {(4 \sin (t))}^{2} \sqrt{16 - {(4 \sin (t))}^{2}} (4 \cos (t)) d t$

Step 2

Simplify terms.

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Step 2.1

Simplify $\sqrt{16 - {(4 \sin (t))}^{2}}$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Apply the product rule to $4 \sin (t)$ .

$\int_{0}^{\frac{π}{2}} {(4 \sin (t))}^{2} \sqrt{16 - (4^{2} \sin^{2} (t))} (4 \cos (t)) d t$

Step 2.1.1.2

Raise $4$ to the power of $2$ .

$\int_{0}^{\frac{π}{2}} {(4 \sin (t))}^{2} \sqrt{16 - (16 \sin^{2} (t))} (4 \cos (t)) d t$

Step 2.1.1.3

Multiply $16$ by $- 1$ .

$\int_{0}^{\frac{π}{2}} {(4 \sin (t))}^{2} \sqrt{16 - 16 \sin^{2} (t)} (4 \cos (t)) d t$

Step 2.1.2

Factor $16$ out of $16$ .

$\int_{0}^{\frac{π}{2}} {(4 \sin (t))}^{2} \sqrt{16 (1) - 16 \sin^{2} (t)} (4 \cos (t)) d t$

Step 2.1.3

Factor $16$ out of $- 16 \sin^{2} (t)$ .

$\int_{0}^{\frac{π}{2}} {(4 \sin (t))}^{2} \sqrt{16 (1) + 16 (- \sin^{2} (t))} (4 \cos (t)) d t$

Step 2.1.4

Factor $16$ out of $16 (1) + 16 (- \sin^{2} (t))$ .

$\int_{0}^{\frac{π}{2}} {(4 \sin (t))}^{2} \sqrt{16 (1 - \sin^{2} (t))} (4 \cos (t)) d t$

Step 2.1.5

Apply pythagorean identity.

$\int_{0}^{\frac{π}{2}} {(4 \sin (t))}^{2} \sqrt{16 \cos^{2} (t)} (4 \cos (t)) d t$

Step 2.1.6

Rewrite $16 \cos^{2} (t)$ as ${(4 \cos (t))}^{2}$ .

$\int_{0}^{\frac{π}{2}} {(4 \sin (t))}^{2} \sqrt{{(4 \cos (t))}^{2}} (4 \cos (t)) d t$

Step 2.1.7

Pull terms out from under the radical, assuming positive real numbers.

$\int_{0}^{\frac{π}{2}} {(4 \sin (t))}^{2} (4 \cos (t)) (4 \cos (t)) d t$

Step 2.2

Simplify.

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Step 2.2.1

Factor $4$ out of $4 \sin (t)$ .

$\int_{0}^{\frac{π}{2}} {(4 (\sin (t)))}^{2} (4 \cos (t)) (4 \cos (t)) d t$

Step 2.2.2

Apply the product rule to $4 (\sin (t))$ .

$\int_{0}^{\frac{π}{2}} 4^{2} \sin^{2} (t) (4 \cos (t)) (4 \cos (t)) d t$

Step 2.2.3

Raise $4$ to the power of $2$ .

$\int_{0}^{\frac{π}{2}} 16 \sin^{2} (t) (4 \cos (t)) (4 \cos (t)) d t$

Step 2.2.4

Multiply $4$ by $16$ .

$\int_{0}^{\frac{π}{2}} 64 \sin^{2} (t) \cos (t) (4 \cos (t)) d t$

Step 2.2.5

Multiply $4$ by $64$ .

$\int_{0}^{\frac{π}{2}} 256 \sin^{2} (t) \cos (t) \cos (t) d t$

Step 2.2.6

Raise $\cos (t)$ to the power of $1$ .

$\int_{0}^{\frac{π}{2}} 256 \sin^{2} (t) (\cos^{1} (t) \cos (t)) d t$

Step 2.2.7

Raise $\cos (t)$ to the power of $1$ .

$\int_{0}^{\frac{π}{2}} 256 \sin^{2} (t) (\cos^{1} (t) \cos^{1} (t)) d t$

Step 2.2.8

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

$\int_{0}^{\frac{π}{2}} 256 \sin^{2} (t) {\cos (t)}^{1 + 1} d t$

Step 2.2.9

Add $1$ and $1$ .

$\int_{0}^{\frac{π}{2}} 256 \sin^{2} (t) \cos^{2} (t) d t$

Step 3

Since $256$ is constant with respect to $t$ , move $256$ out of the integral.

$256 \int_{0}^{\frac{π}{2}} \sin^{2} (t) \cos^{2} (t) d t$

Step 4

Use the half-angle formula to rewrite $\cos^{2} (t)$ as $\frac{1 + \cos (2 t)}{2}$ .

$256 \int_{0}^{\frac{π}{2}} \sin^{2} (t) \frac{1 + \cos (2 t)}{2} d t$

Step 5

Use the half-angle formula to rewrite $\sin^{2} (t)$ as $\frac{1 - \cos (2 t)}{2}$ .

$256 \int_{0}^{\frac{π}{2}} \frac{1 - \cos (2 t)}{2} \cdot \frac{1 + \cos (2 t)}{2} d t$

Step 6

Simplify.

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Step 6.1

Multiply $\frac{1 - \cos (2 t)}{2}$ by $\frac{1 + \cos (2 t)}{2}$ .

$256 \int_{0}^{\frac{π}{2}} \frac{(1 - \cos (2 t)) (1 + \cos (2 t))}{2 \cdot 2} d t$

Step 6.2

Multiply $2$ by $2$ .

$256 \int_{0}^{\frac{π}{2}} \frac{(1 - \cos (2 t)) (1 + \cos (2 t))}{4} d t$

Step 7

Since $\frac{1}{4}$ is constant with respect to $t$ , move $\frac{1}{4}$ out of the integral.

$256 (\frac{1}{4} \int_{0}^{\frac{π}{2}} (1 - \cos (2 t)) (1 + \cos (2 t)) d t)$

Step 8

Simplify.

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Step 8.1

Combine $\frac{1}{4}$ and $256$ .

$\frac{256}{4} \int_{0}^{\frac{π}{2}} (1 - \cos (2 t)) (1 + \cos (2 t)) d t$

Step 8.2

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

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Step 8.2.1

Factor $4$ out of $256$ .

$\frac{4 \cdot 64}{4} \int_{0}^{\frac{π}{2}} (1 - \cos (2 t)) (1 + \cos (2 t)) d t$

Step 8.2.2

Cancel the common factors.

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Step 8.2.2.1

Factor $4$ out of $4$ .

$\frac{4 \cdot 64}{4 (1)} \int_{0}^{\frac{π}{2}} (1 - \cos (2 t)) (1 + \cos (2 t)) d t$

Step 8.2.2.2

Cancel the common factor.

$\frac{4 \cdot 64}{4 \cdot 1} \int_{0}^{\frac{π}{2}} (1 - \cos (2 t)) (1 + \cos (2 t)) d t$

Step 8.2.2.3

Rewrite the expression.

$\frac{64}{1} \int_{0}^{\frac{π}{2}} (1 - \cos (2 t)) (1 + \cos (2 t)) d t$

Step 8.2.2.4

Divide $64$ by $1$ .

$64 \int_{0}^{\frac{π}{2}} (1 - \cos (2 t)) (1 + \cos (2 t)) d t$

Step 9

Let $u_{1} = 2 t$ . Then $d u_{1} = 2 d t$ , so $\frac{1}{2} d u_{1} = d t$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 9.1

Let $u_{1} = 2 t$ . Find $\frac{d u_{1}}{d t}$ .

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Step 9.1.1

Differentiate $2 t$ .

$\frac{d}{d t} [2 t]$

Step 9.1.2

Since $2$ is constant with respect to $t$ , the derivative of $2 t$ with respect to $t$ is $2 \frac{d}{d t} [t]$ .

$2 \frac{d}{d t} [t]$

Step 9.1.3

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 1$ .

$2 \cdot 1$

Step 9.1.4

Multiply $2$ by $1$ .

$2$

Step 9.2

Substitute the lower limit in for $t$ in $u_{1} = 2 t$ .

$u_{lower} = 2 \cdot 0$

Step 9.3

Multiply $2$ by $0$ .

$u_{lower} = 0$

Step 9.4

Substitute the upper limit in for $t$ in $u_{1} = 2 t$ .

$u_{upper} = 2 \frac{π}{2}$

Step 9.5

Cancel the common factor of $2$ .

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Step 9.5.1

Cancel the common factor.

$u_{upper} = 2 \frac{π}{2}$

Step 9.5.2

Rewrite the expression.

$u_{upper} = π$

Step 9.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 0$

$u_{upper} = π$

Step 9.7

Rewrite the problem using $u_{1}$ , $d u_{1}$ , and the new limits of integration.

$64 \int_{0}^{π} (1 - \cos (u_{1})) (1 + \cos (u_{1})) \frac{1}{2} d u_{1}$

Step 10

Since $\frac{1}{2}$ is constant with respect to $u_{1}$ , move $\frac{1}{2}$ out of the integral.

$64 (\frac{1}{2} \int_{0}^{π} (1 - \cos (u_{1})) (1 + \cos (u_{1})) d u_{1})$

Step 11

Simplify by multiplying through.

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Step 11.1

Simplify.

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Step 11.1.1

Combine $\frac{1}{2}$ and $64$ .

$\frac{64}{2} \int_{0}^{π} (1 - \cos (u_{1})) (1 + \cos (u_{1})) d u_{1}$

Step 11.1.2

Cancel the common factor of $64$ and $2$ .

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Step 11.1.2.1

Factor $2$ out of $64$ .

$\frac{2 \cdot 32}{2} \int_{0}^{π} (1 - \cos (u_{1})) (1 + \cos (u_{1})) d u_{1}$

Step 11.1.2.2

Cancel the common factors.

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Step 11.1.2.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 32}{2 (1)} \int_{0}^{π} (1 - \cos (u_{1})) (1 + \cos (u_{1})) d u_{1}$

Step 11.1.2.2.2

Cancel the common factor.

$\frac{2 \cdot 32}{2 \cdot 1} \int_{0}^{π} (1 - \cos (u_{1})) (1 + \cos (u_{1})) d u_{1}$

Step 11.1.2.2.3

Rewrite the expression.

$\frac{32}{1} \int_{0}^{π} (1 - \cos (u_{1})) (1 + \cos (u_{1})) d u_{1}$

Step 11.1.2.2.4

Divide $32$ by $1$ .

$32 \int_{0}^{π} (1 - \cos (u_{1})) (1 + \cos (u_{1})) d u_{1}$

Step 11.2

Expand $(1 - \cos (u_{1})) (1 + \cos (u_{1}))$ .

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Step 11.2.1

Apply the distributive property.

$32 \int_{0}^{π} 1 (1 + \cos (u_{1})) - \cos (u_{1}) (1 + \cos (u_{1})) d u_{1}$

Step 11.2.2

Apply the distributive property.

$32 \int_{0}^{π} 1 \cdot 1 + 1 \cos (u_{1}) - \cos (u_{1}) (1 + \cos (u_{1})) d u_{1}$

Step 11.2.3

Apply the distributive property.

$32 \int_{0}^{π} 1 \cdot 1 + 1 \cos (u_{1}) - \cos (u_{1}) \cdot 1 - \cos (u_{1}) \cos (u_{1}) d u_{1}$

Step 11.2.4

Move $\cos (u_{1})$ .

$32 \int_{0}^{π} 1 \cdot 1 + 1 \cos (u_{1}) - 1 \cdot 1 \cos (u_{1}) - \cos (u_{1}) \cos (u_{1}) d u_{1}$

Step 11.2.5

Multiply $1$ by $1$ .

$32 \int_{0}^{π} 1 + 1 \cos (u_{1}) - 1 \cdot 1 \cos (u_{1}) - \cos (u_{1}) \cos (u_{1}) d u_{1}$

Step 11.2.6

Multiply $\cos (u_{1})$ by $1$ .

$32 \int_{0}^{π} 1 + \cos (u_{1}) - 1 \cdot 1 \cos (u_{1}) - \cos (u_{1}) \cos (u_{1}) d u_{1}$

Step 11.2.7

Multiply $- 1$ by $1$ .

$32 \int_{0}^{π} 1 + \cos (u_{1}) - \cos (u_{1}) - \cos (u_{1}) \cos (u_{1}) d u_{1}$

Step 11.2.8

Factor out negative.

$32 \int_{0}^{π} 1 + \cos (u_{1}) - \cos (u_{1}) - (\cos (u_{1}) \cos (u_{1})) d u_{1}$

Step 11.2.9

Raise $\cos (u_{1})$ to the power of $1$ .

$32 \int_{0}^{π} 1 + \cos (u_{1}) - \cos (u_{1}) - (\cos^{1} (u_{1}) \cos (u_{1})) d u_{1}$

Step 11.2.10

Raise $\cos (u_{1})$ to the power of $1$ .

$32 \int_{0}^{π} 1 + \cos (u_{1}) - \cos (u_{1}) - (\cos^{1} (u_{1}) \cos^{1} (u_{1})) d u_{1}$

Step 11.2.11

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

$32 \int_{0}^{π} 1 + \cos (u_{1}) - \cos (u_{1}) - {\cos (u_{1})}^{1 + 1} d u_{1}$

Step 11.2.12

Add $1$ and $1$ .

$32 \int_{0}^{π} 1 + \cos (u_{1}) - \cos (u_{1}) - \cos^{2} (u_{1}) d u_{1}$

Step 11.2.13

Subtract $\cos (u_{1})$ from $\cos (u_{1})$ .

$32 \int_{0}^{π} 1 + 0 - \cos^{2} (u_{1}) d u_{1}$

Step 11.2.14

Subtract $\cos^{2} (u_{1})$ from $0$ .

$32 \int_{0}^{π} 1 - \cos^{2} (u_{1}) d u_{1}$

Step 12

Split the single integral into multiple integrals.

$32 (\int_{0}^{π} d u_{1} + \int_{0}^{π} - \cos^{2} (u_{1}) d u_{1})$

Step 13

Apply the constant rule.

$32 (u_{1}]_{0}^{π} + \int_{0}^{π} - \cos^{2} (u_{1}) d u_{1})$

Step 14

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

$32 (u_{1}]_{0}^{π} - \int_{0}^{π} \cos^{2} (u_{1}) d u_{1})$

Step 15

Use the half-angle formula to rewrite $\cos^{2} (u_{1})$ as $\frac{1 + \cos (2 u_{1})}{2}$ .

$32 (u_{1}]_{0}^{π} - \int_{0}^{π} \frac{1 + \cos (2 u_{1})}{2} d u_{1})$

Step 16

Since $\frac{1}{2}$ is constant with respect to $u_{1}$ , move $\frac{1}{2}$ out of the integral.

$32 (u_{1}]_{0}^{π} - (\frac{1}{2} \int_{0}^{π} 1 + \cos (2 u_{1}) d u_{1}))$

Step 17

Split the single integral into multiple integrals.

$32 (u_{1}]_{0}^{π} - \frac{1}{2} (\int_{0}^{π} d u_{1} + \int_{0}^{π} \cos (2 u_{1}) d u_{1}))$

Step 18

Apply the constant rule.

$32 (u_{1}]_{0}^{π} - \frac{1}{2} (u_{1}]_{0}^{π} + \int_{0}^{π} \cos (2 u_{1}) d u_{1}))$

Step 19

Let $u_{2} = 2 u_{1}$ . Then $d u_{2} = 2 d u_{1}$ , so $\frac{1}{2} d u_{2} = d u_{1}$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 19.1

Let $u_{2} = 2 u_{1}$ . Find $\frac{d u_{2}}{d u_{1}}$ .

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Step 19.1.1

Differentiate $2 u_{1}$ .

$\frac{d}{d u_{1}} [2 u_{1}]$

Step 19.1.2

Since $2$ is constant with respect to $u_{1}$ , the derivative of $2 u_{1}$ with respect to $u_{1}$ is $2 \frac{d}{d u_{1}} [u_{1}]$ .

$2 \frac{d}{d u_{1}} [u_{1}]$

Step 19.1.3

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 1$ .

$2 \cdot 1$

Step 19.1.4

Multiply $2$ by $1$ .

$2$

Step 19.2

Substitute the lower limit in for $u_{1}$ in $u_{2} = 2 u_{1}$ .

$u_{lower} = 2 \cdot 0$

Step 19.3

Multiply $2$ by $0$ .

$u_{lower} = 0$

Step 19.4

Substitute the upper limit in for $u_{1}$ in $u_{2} = 2 u_{1}$ .

$u_{upper} = 2 π$

Step 19.5

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 0$

$u_{upper} = 2 π$

Step 19.6

Rewrite the problem using $u_{2}$ , $d u_{2}$ , and the new limits of integration.

$32 (u_{1}]_{0}^{π} - \frac{1}{2} (u_{1}]_{0}^{π} + \int_{0}^{2 π} \cos (u_{2}) \frac{1}{2} d u_{2}))$

Step 20

Combine $\cos (u_{2})$ and $\frac{1}{2}$ .

$32 (u_{1}]_{0}^{π} - \frac{1}{2} (u_{1}]_{0}^{π} + \int_{0}^{2 π} \frac{\cos (u_{2})}{2} d u_{2}))$

Step 21

Since $\frac{1}{2}$ is constant with respect to $u_{2}$ , move $\frac{1}{2}$ out of the integral.

$32 (u_{1}]_{0}^{π} - \frac{1}{2} (u_{1}]_{0}^{π} + \frac{1}{2} \int_{0}^{2 π} \cos (u_{2}) d u_{2}))$

Step 22

The integral of $\cos (u_{2})$ with respect to $u_{2}$ is $\sin (u_{2})$ .

$32 (u_{1}]_{0}^{π} - \frac{1}{2} (u_{1}]_{0}^{π} + \frac{1}{2} \sin (u_{2})]_{0}^{2 π}))$

Step 23

Combine $\frac{1}{2}$ and $\sin (u_{2})]_{0}^{2 π}$ .

$32 (u_{1}]_{0}^{π} - \frac{1}{2} (u_{1}]_{0}^{π} + \frac{\sin (u_{2})]_{0}^{2 π}}{2}))$

Step 24

Substitute and simplify.

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Step 24.1

Evaluate $u_{1}$ at $π$ and at $0$ .

$32 ((π) + 0 - \frac{1}{2} (u_{1}]_{0}^{π} + \frac{\sin (u_{2})]_{0}^{2 π}}{2}))$

Step 24.2

Evaluate $u_{1}$ at $π$ and at $0$ .

$32 (π + 0 - \frac{1}{2} (π + 0 + \frac{\sin (u_{2})]_{0}^{2 π}}{2}))$

Step 24.3

Evaluate $\sin (u_{2})$ at $2 π$ and at $0$ .

$32 (π + 0 - \frac{1}{2} (π + 0 + \frac{(\sin (2 π)) - \sin (0)}{2}))$

Step 24.4

Simplify.

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Step 24.4.1

Add $π$ and $0$ .

$32 (π - \frac{1}{2} (π + 0 + \frac{(\sin (2 π)) - \sin (0)}{2}))$

Step 24.4.2

Add $π$ and $0$ .

$32 (π - \frac{1}{2} (π + \frac{\sin (2 π) - \sin (0)}{2}))$

Step 25

Simplify.

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Step 25.1

The exact value of $\sin (0)$ is $0$ .

$32 (π - \frac{1}{2} (π + \frac{\sin (2 π) - 0}{2}))$

Step 25.2

Multiply $- 1$ by $0$ .

$32 (π - \frac{1}{2} (π + \frac{\sin (2 π) + 0}{2}))$

Step 25.3

Add $\sin (2 π)$ and $0$ .

$32 (π - \frac{1}{2} (π + \frac{\sin (2 π)}{2}))$

Step 26

Simplify.

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Step 26.1

Simplify each term.

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Step 26.1.1

Simplify the numerator.

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Step 26.1.1.1

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$32 (π - \frac{1}{2} (π + \frac{\sin (0)}{2}))$

Step 26.1.1.2

The exact value of $\sin (0)$ is $0$ .

$32 (π - \frac{1}{2} (π + \frac{0}{2}))$

Step 26.1.2

Divide $0$ by $2$ .

$32 (π - \frac{1}{2} (π + 0))$

Step 26.2

Add $π$ and $0$ .

$32 (π - \frac{1}{2} π)$

Step 26.3

Combine $π$ and $\frac{1}{2}$ .

$32 (π - \frac{π}{2})$

Step 26.4

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

$32 (π \cdot \frac{2}{2} - \frac{π}{2})$

Step 26.5

Combine $π$ and $\frac{2}{2}$ .

$32 (\frac{π \cdot 2}{2} - \frac{π}{2})$

Step 26.6

Combine the numerators over the common denominator.

$32 \frac{π \cdot 2 - π}{2}$

Step 26.7

Move $2$ to the left of $π$ .

$32 \frac{2 \cdot π - π}{2}$

Step 26.8

Cancel the common factor of $2$ .

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Step 26.8.1

Factor $2$ out of $32$ .

$2 (16) \frac{2 π - π}{2}$

Step 26.8.2

Cancel the common factor.

$2 \cdot 16 \frac{2 π - π}{2}$

Step 26.8.3

Rewrite the expression.

$16 (2 π - π)$

Step 26.9

Subtract $π$ from $2 π$ .

$16 π$

Step 27

The result can be shown in multiple forms.

Exact Form:

$16 π$

Decimal Form:

$50.26548245 \dots$

Step 28