Calculus Examples

$\int_{0}^{12} \sqrt{36 - {(x - 6)}^{2}} d x$

Step 1

Let $u_{1} = x - 6$ . Then $d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 1.1

Let $u_{1} = x - 6$ . Find $\frac{d u_{1}}{d x}$ .

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Step 1.1.1

Differentiate $x - 6$ .

$\frac{d}{d x} [x - 6]$

Step 1.1.2

By the Sum Rule, the derivative of $x - 6$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 6]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- 6]$

Step 1.1.3

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

$1 + \frac{d}{d x} [- 6]$

Step 1.1.4

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

$1 + 0$

Step 1.1.5

Add $1$ and $0$ .

$1$

Step 1.2

Substitute the lower limit in for $x$ in $u_{1} = x - 6$ .

$u_{lower} = 0 - 6$

Step 1.3

Subtract $6$ from $0$ .

$u_{lower} = - 6$

Step 1.4

Substitute the upper limit in for $x$ in $u_{1} = x - 6$ .

$u_{upper} = 12 - 6$

Step 1.5

Subtract $6$ from $12$ .

$u_{upper} = 6$

Step 1.6

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

$u_{lower} = - 6$

$u_{upper} = 6$

Step 1.7

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

$\int_{- 6}^{6} \sqrt{36 - {u_{1}}^{2}} d u_{1}$

Step 2

Let $u_{1} = 6 \sin (t)$ , where $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then $d u_{1} = 6 \cos (t) d t$ . Note that since $- \frac{π}{2} \leq t \leq \frac{π}{2}$ , $6 \cos (t)$ is positive.

$\int_{- \frac{π}{2}}^{\frac{π}{2}} \sqrt{36 - {(6 \sin (t))}^{2}} (6 \cos (t)) d t$

Step 3

Simplify terms.

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Step 3.1

Simplify $\sqrt{36 - {(6 \sin (t))}^{2}}$ .

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Step 3.1.1

Simplify each term.

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Step 3.1.1.1

Apply the product rule to $6 \sin (t)$ .

$\int_{- \frac{π}{2}}^{\frac{π}{2}} \sqrt{36 - (6^{2} \sin^{2} (t))} (6 \cos (t)) d t$

Step 3.1.1.2

Raise $6$ to the power of $2$ .

$\int_{- \frac{π}{2}}^{\frac{π}{2}} \sqrt{36 - (36 \sin^{2} (t))} (6 \cos (t)) d t$

Step 3.1.1.3

Multiply $36$ by $- 1$ .

$\int_{- \frac{π}{2}}^{\frac{π}{2}} \sqrt{36 - 36 \sin^{2} (t)} (6 \cos (t)) d t$

Step 3.1.2

Factor $36$ out of $36$ .

$\int_{- \frac{π}{2}}^{\frac{π}{2}} \sqrt{36 (1) - 36 \sin^{2} (t)} (6 \cos (t)) d t$

Step 3.1.3

Factor $36$ out of $- 36 \sin^{2} (t)$ .

$\int_{- \frac{π}{2}}^{\frac{π}{2}} \sqrt{36 (1) + 36 (- \sin^{2} (t))} (6 \cos (t)) d t$

Step 3.1.4

Factor $36$ out of $36 (1) + 36 (- \sin^{2} (t))$ .

$\int_{- \frac{π}{2}}^{\frac{π}{2}} \sqrt{36 (1 - \sin^{2} (t))} (6 \cos (t)) d t$

Step 3.1.5

Apply pythagorean identity.

$\int_{- \frac{π}{2}}^{\frac{π}{2}} \sqrt{36 \cos^{2} (t)} (6 \cos (t)) d t$

Step 3.1.6

Rewrite $36 \cos^{2} (t)$ as ${(6 \cos (t))}^{2}$ .

$\int_{- \frac{π}{2}}^{\frac{π}{2}} \sqrt{{(6 \cos (t))}^{2}} (6 \cos (t)) d t$

Step 3.1.7

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

$\int_{- \frac{π}{2}}^{\frac{π}{2}} 6 \cos (t) (6 \cos (t)) d t$

Step 3.2

Simplify.

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Step 3.2.1

Multiply $6$ by $6$ .

$\int_{- \frac{π}{2}}^{\frac{π}{2}} 36 \cos (t) \cos (t) d t$

Step 3.2.2

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

$\int_{- \frac{π}{2}}^{\frac{π}{2}} 36 (\cos^{1} (t) \cos (t)) d t$

Step 3.2.3

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

$\int_{- \frac{π}{2}}^{\frac{π}{2}} 36 (\cos^{1} (t) \cos^{1} (t)) d t$

Step 3.2.4

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

$\int_{- \frac{π}{2}}^{\frac{π}{2}} 36 {\cos (t)}^{1 + 1} d t$

Step 3.2.5

Add $1$ and $1$ .

$\int_{- \frac{π}{2}}^{\frac{π}{2}} 36 \cos^{2} (t) d t$

Step 4

Since $36$ is constant with respect to $t$ , move $36$ out of the integral.

$36 \int_{- \frac{π}{2}}^{\frac{π}{2}} \cos^{2} (t) d t$

Step 5

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

$36 \int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{1 + \cos (2 t)}{2} d t$

Step 6

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

$36 (\frac{1}{2} \int_{- \frac{π}{2}}^{\frac{π}{2}} 1 + \cos (2 t) d t)$

Step 7

Simplify.

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Step 7.1

Combine $\frac{1}{2}$ and $36$ .

$\frac{36}{2} \int_{- \frac{π}{2}}^{\frac{π}{2}} 1 + \cos (2 t) d t$

Step 7.2

Cancel the common factor of $36$ and $2$ .

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Step 7.2.1

Factor $2$ out of $36$ .

$\frac{2 \cdot 18}{2} \int_{- \frac{π}{2}}^{\frac{π}{2}} 1 + \cos (2 t) d t$

Step 7.2.2

Cancel the common factors.

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Step 7.2.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 18}{2 (1)} \int_{- \frac{π}{2}}^{\frac{π}{2}} 1 + \cos (2 t) d t$

Step 7.2.2.2

Cancel the common factor.

$\frac{2 \cdot 18}{2 \cdot 1} \int_{- \frac{π}{2}}^{\frac{π}{2}} 1 + \cos (2 t) d t$

Step 7.2.2.3

Rewrite the expression.

$\frac{18}{1} \int_{- \frac{π}{2}}^{\frac{π}{2}} 1 + \cos (2 t) d t$

Step 7.2.2.4

Divide $18$ by $1$ .

$18 \int_{- \frac{π}{2}}^{\frac{π}{2}} 1 + \cos (2 t) d t$

Step 8

Split the single integral into multiple integrals.

$18 (\int_{- \frac{π}{2}}^{\frac{π}{2}} d t + \int_{- \frac{π}{2}}^{\frac{π}{2}} \cos (2 t) d t)$

Step 9

Apply the constant rule.

$18 (t]_{- \frac{π}{2}}^{\frac{π}{2}} + \int_{- \frac{π}{2}}^{\frac{π}{2}} \cos (2 t) d t)$

Step 10

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

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Step 10.1

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

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Step 10.1.1

Differentiate $2 t$ .

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

Step 10.1.2

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 10.1.3

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 10.1.4

Multiply $2$ by $1$ .

$2$

Step 10.2

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

$u_{lower} = 2 (- \frac{π}{2})$

Step 10.3

Cancel the common factor of $2$ .

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Step 10.3.1

Move the leading negative in $- \frac{π}{2}$ into the numerator.

$u_{lower} = 2 \frac{- π}{2}$

Step 10.3.2

Cancel the common factor.

$u_{lower} = 2 \frac{- π}{2}$

Step 10.3.3

Rewrite the expression.

$u_{lower} = - π$

Step 10.4

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

$u_{upper} = 2 \frac{π}{2}$

Step 10.5

Cancel the common factor of $2$ .

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Step 10.5.1

Cancel the common factor.

$u_{upper} = 2 \frac{π}{2}$

Step 10.5.2

Rewrite the expression.

$u_{upper} = π$

Step 10.6

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

$u_{lower} = - π$

$u_{upper} = π$

Step 10.7

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

$18 (t]_{- \frac{π}{2}}^{\frac{π}{2}} + \int_{- π}^{π} \cos (u_{2}) \frac{1}{2} d u_{2})$

Step 11

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

$18 (t]_{- \frac{π}{2}}^{\frac{π}{2}} + \int_{- π}^{π} \frac{\cos (u_{2})}{2} d u_{2})$

Step 12

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

$18 (t]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{1}{2} \int_{- π}^{π} \cos (u_{2}) d u_{2})$

Step 13

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

$18 (t]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{1}{2} \sin (u_{2})]_{- π}^{π})$

Step 14

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

$18 (t]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{\sin (u_{2})]_{- π}^{π}}{2})$

Step 15

Substitute and simplify.

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Step 15.1

Evaluate $t$ at $\frac{π}{2}$ and at $- \frac{π}{2}$ .

$18 ((\frac{π}{2}) + \frac{π}{2} + \frac{\sin (u_{2})]_{- π}^{π}}{2})$

Step 15.2

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

$18 ((\frac{π}{2}) + \frac{π}{2} + \frac{(\sin (π)) - \sin (- π)}{2})$

Step 15.3

Simplify.

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Step 15.3.1

Combine the numerators over the common denominator.

$18 (\frac{π + π}{2} + \frac{(\sin (π)) - \sin (- π)}{2})$

Step 15.3.2

Add $π$ and $π$ .

$18 (\frac{2 π}{2} + \frac{(\sin (π)) - \sin (- π)}{2})$

Step 15.3.3

Cancel the common factor of $2$ .

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Step 15.3.3.1

Cancel the common factor.

$18 (\frac{2 π}{2} + \frac{(\sin (π)) - \sin (- π)}{2})$

Step 15.3.3.2

Divide $π$ by $1$ .

$18 (π + \frac{(\sin (π)) - \sin (- π)}{2})$

$18 (π + \frac{\sin (π) - \sin (- π)}{2})$

Step 16

Simplify.

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Step 16.1

Simplify the numerator.

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Step 16.1.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$18 (π + \frac{\sin (0) - \sin (- π)}{2})$

Step 16.1.2

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

$18 (π + \frac{0 - \sin (- π)}{2})$

Step 16.1.3

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$18 (π + \frac{0 - \sin (0)}{2})$

Step 16.1.4

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

$18 (π + \frac{0 - 0}{2})$

Step 16.1.5

Multiply $- 1$ by $0$ .

$18 (π + \frac{0 + 0}{2})$

Step 16.1.6

Add $0$ and $0$ .

$18 (π + \frac{0}{2})$

Step 16.2

Divide $0$ by $2$ .

$18 (π + 0)$

Step 17

Add $π$ and $0$ .

$18 π$

Step 18

The result can be shown in multiple forms.

Exact Form:

$18 π$

Decimal Form:

$56.54866776 \dots$

Step 19