Calculus Examples

$\int_{- 2}^{1} \sqrt{3^{2} - x^{2}} d x$

Step 1

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

$\int_{- 2}^{1} \sqrt{9 - x^{2}} d x$

Step 2

Let $x = 3 \sin (t)$ , where $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then $d x = 3 \cos (t) d t$ . Note that since $- \frac{π}{2} \leq t \leq \frac{π}{2}$ , $3 \cos (t)$ is positive.

$\int_{- 0.72972765}^{0.3398369} \sqrt{9 - {(3 \sin (t))}^{2}} (3 \cos (t)) d t$

Step 3

Simplify terms.

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Step 3.1

Simplify $\sqrt{9 - {(3 \sin (t))}^{2}}$ .

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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 $3 \sin (t)$ .

$\int_{- 0.72972765}^{0.3398369} \sqrt{9 - (3^{2} \sin^{2} (t))} (3 \cos (t)) d t$

Step 3.1.1.2

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

$\int_{- 0.72972765}^{0.3398369} \sqrt{9 - (9 \sin^{2} (t))} (3 \cos (t)) d t$

Step 3.1.1.3

Multiply $9$ by $- 1$ .

$\int_{- 0.72972765}^{0.3398369} \sqrt{9 - 9 \sin^{2} (t)} (3 \cos (t)) d t$

Step 3.1.2

Factor $9$ out of $9$ .

$\int_{- 0.72972765}^{0.3398369} \sqrt{9 (1) - 9 \sin^{2} (t)} (3 \cos (t)) d t$

Step 3.1.3

Factor $9$ out of $- 9 \sin^{2} (t)$ .

$\int_{- 0.72972765}^{0.3398369} \sqrt{9 (1) + 9 (- \sin^{2} (t))} (3 \cos (t)) d t$

Step 3.1.4

Factor $9$ out of $9 (1) + 9 (- \sin^{2} (t))$ .

$\int_{- 0.72972765}^{0.3398369} \sqrt{9 (1 - \sin^{2} (t))} (3 \cos (t)) d t$

Step 3.1.5

Apply pythagorean identity.

$\int_{- 0.72972765}^{0.3398369} \sqrt{9 \cos^{2} (t)} (3 \cos (t)) d t$

Step 3.1.6

Rewrite $9 \cos^{2} (t)$ as ${(3 \cos (t))}^{2}$ .

$\int_{- 0.72972765}^{0.3398369} \sqrt{{(3 \cos (t))}^{2}} (3 \cos (t)) d t$

Step 3.1.7

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

$\int_{- 0.72972765}^{0.3398369} 3 \cos (t) (3 \cos (t)) d t$

Step 3.2

Simplify.

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Step 3.2.1

Multiply $3$ by $3$ .

$\int_{- 0.72972765}^{0.3398369} 9 \cos (t) \cos (t) d t$

Step 3.2.2

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

$\int_{- 0.72972765}^{0.3398369} 9 (\cos^{1} (t) \cos (t)) d t$

Step 3.2.3

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

$\int_{- 0.72972765}^{0.3398369} 9 (\cos^{1} (t) \cos^{1} (t)) d t$

Step 3.2.4

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

$\int_{- 0.72972765}^{0.3398369} 9 {\cos (t)}^{1 + 1} d t$

Step 3.2.5

Add $1$ and $1$ .

$\int_{- 0.72972765}^{0.3398369} 9 \cos^{2} (t) d t$

Step 4

Since $9$ is constant with respect to $t$ , move $9$ out of the integral.

$9 \int_{- 0.72972765}^{0.3398369} \cos^{2} (t) d t$

Step 5

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

$9 \int_{- 0.72972765}^{0.3398369} \frac{1 + \cos (2 t)}{2} d t$

Step 6

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

$9 (\frac{1}{2} \int_{- 0.72972765}^{0.3398369} 1 + \cos (2 t) d t)$

Step 7

Combine $\frac{1}{2}$ and $9$ .

$\frac{9}{2} \int_{- 0.72972765}^{0.3398369} 1 + \cos (2 t) d t$

Step 8

Split the single integral into multiple integrals.

$\frac{9}{2} (\int_{- 0.72972765}^{0.3398369} d t + \int_{- 0.72972765}^{0.3398369} \cos (2 t) d t)$

Step 9

Apply the constant rule.

$\frac{9}{2} (t]_{- 0.72972765}^{0.3398369} + \int_{- 0.72972765}^{0.3398369} \cos (2 t) d t)$

Step 10

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

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Step 10.1

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

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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 $u = 2 t$ .

$u_{lower} = 2 \cdot - 0.72972765$

Step 10.3

Multiply $2$ by $- 0.72972765$ .

$u_{lower} = - 1.45945531$

Step 10.4

Substitute the upper limit in for $t$ in $u = 2 t$ .

$u_{upper} = 2 \cdot 0.3398369$

Step 10.5

Multiply $2$ by $0.3398369$ .

$u_{upper} = 0.67967381$

Step 10.6

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

$u_{lower} = - 1.45945531$

$u_{upper} = 0.67967381$

Step 10.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$\frac{9}{2} (t]_{- 0.72972765}^{0.3398369} + \int_{- 1.45945531}^{0.67967381} \cos (u) \frac{1}{2} d u)$

Step 11

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

$\frac{9}{2} (t]_{- 0.72972765}^{0.3398369} + \int_{- 1.45945531}^{0.67967381} \frac{\cos (u)}{2} d u)$

Step 12

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

$\frac{9}{2} (t]_{- 0.72972765}^{0.3398369} + \frac{1}{2} \int_{- 1.45945531}^{0.67967381} \cos (u) d u)$

Step 13

The integral of $\cos (u)$ with respect to $u$ is $\sin (u)$ .

$\frac{9}{2} (t]_{- 0.72972765}^{0.3398369} + \frac{1}{2} \sin (u)]_{- 1.45945531}^{0.67967381})$

Step 14

Substitute and simplify.

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Step 14.1

Evaluate $t$ at $0.3398369$ and at $- 0.72972765$ .

$\frac{9}{2} ((0.3398369) + 0.72972765 + \frac{1}{2} \sin (u)]_{- 1.45945531}^{0.67967381})$

Step 14.2

Evaluate $\sin (u)$ at $0.67967381$ and at $- 1.45945531$ .

$\frac{9}{2} ((0.3398369) + 0.72972765 + \frac{1}{2} ((\sin (0.67967381)) - \sin (- 1.45945531)))$

Step 14.3

Add $0.3398369$ and $0.72972765$ .

$\frac{9}{2} (1.06956456 + \frac{1}{2} (\sin (0.67967381) - \sin (- 1.45945531)))$

Step 15

Simplify.

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Step 15.1

Simplify each term.

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Step 15.1.1

Apply the distributive property.

$\frac{9}{2} (1.06956456 + \frac{1}{2} \sin (0.67967381) + \frac{1}{2} (- \sin (- 1.45945531)))$

Step 15.1.2

Combine $\frac{1}{2}$ and $\sin (0.67967381)$ .

$\frac{9}{2} (1.06956456 + \frac{\sin (0.67967381)}{2} + \frac{1}{2} (- \sin (- 1.45945531)))$

Step 15.1.3

Combine $\frac{1}{2}$ and $\sin (- 1.45945531)$ .

$\frac{9}{2} (1.06956456 + \frac{\sin (0.67967381)}{2} - \frac{\sin (- 1.45945531)}{2})$

Step 15.1.4

Simplify each term.

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Step 15.1.4.1

Evaluate $\sin (0.67967381)$ .

$\frac{9}{2} (1.06956456 + \frac{0.62853936}{2} - \frac{\sin (- 1.45945531)}{2})$

Step 15.1.4.2

Divide $0.62853936$ by $2$ .

$\frac{9}{2} (1.06956456 + 0.31426968 - \frac{\sin (- 1.45945531)}{2})$

Step 15.1.4.3

Evaluate $\sin (- 1.45945531)$ .

$\frac{9}{2} (1.06956456 + 0.31426968 - \frac{- 0.99380799}{2})$

Step 15.1.4.4

Divide $- 0.99380799$ by $2$ .

$\frac{9}{2} (1.06956456 + 0.31426968 - - 0.49690399)$

Step 15.1.4.5

Multiply $- 1$ by $- 0.49690399$ .

$\frac{9}{2} (1.06956456 + 0.31426968 + 0.49690399)$

Step 15.1.5

Add $0.31426968$ and $0.49690399$ .

$\frac{9}{2} (1.06956456 + 0.81117367)$

Step 15.2

Add $1.06956456$ and $0.81117367$ .

$\frac{9}{2} \cdot 1.88073824$

Step 15.3

Multiply $\frac{9}{2} \cdot 1.88073824$ .

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Step 15.3.1

Combine $\frac{9}{2}$ and $1.88073824$ .

$\frac{9 \cdot 1.88073824}{2}$

Step 15.3.2

Multiply $9$ by $1.88073824$ .

$\frac{16.92664417}{2}$

Step 15.4

Divide $16.92664417$ by $2$ .

$8.46332208$

Step 16