Calculus Examples

$\int_{- 8}^{0} (\frac{y}{8} + \sqrt{y + 9}) d y$

Step 1

Remove parentheses.

$\int_{- 8}^{0} \frac{y}{8} + \sqrt{y + 9} d y$

Step 2

Split the single integral into multiple integrals.

$\int_{- 8}^{0} \frac{y}{8} d y + \int_{- 8}^{0} \sqrt{y + 9} d y$

Step 3

Since $\frac{1}{8}$ is constant with respect to $y$ , move $\frac{1}{8}$ out of the integral.

$\frac{1}{8} \int_{- 8}^{0} y d y + \int_{- 8}^{0} \sqrt{y + 9} d y$

Step 4

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

$\frac{1}{8} \frac{1}{2} y^{2}]_{- 8}^{0} + \int_{- 8}^{0} \sqrt{y + 9} d y$

Step 5

Let $u = y + 9$ . Then $d u = d y$ . Rewrite using $u$ and $d$ $u$ .

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Step 5.1

Let $u = y + 9$ . Find $\frac{d u}{d y}$ .

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Step 5.1.1

Differentiate $y + 9$ .

$\frac{d}{d y} [y + 9]$

Step 5.1.2

By the Sum Rule, the derivative of $y + 9$ with respect to $y$ is $\frac{d}{d y} [y] + \frac{d}{d y} [9]$ .

$\frac{d}{d y} [y] + \frac{d}{d y} [9]$

Step 5.1.3

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

$1 + \frac{d}{d y} [9]$

Step 5.1.4

Since $9$ is constant with respect to $y$ , the derivative of $9$ with respect to $y$ is $0$ .

$1 + 0$

Step 5.1.5

Add $1$ and $0$ .

$1$

Step 5.2

Substitute the lower limit in for $y$ in $u = y + 9$ .

$u_{lower} = - 8 + 9$

Step 5.3

Add $- 8$ and $9$ .

$u_{lower} = 1$

Step 5.4

Substitute the upper limit in for $y$ in $u = y + 9$ .

$u_{upper} = 0 + 9$

Step 5.5

Add $0$ and $9$ .

$u_{upper} = 9$

Step 5.6

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

$u_{lower} = 1$

$u_{upper} = 9$

Step 5.7

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

$\frac{1}{8} \frac{1}{2} y^{2}]_{- 8}^{0} + \int_{1}^{9} \sqrt{u} d u$

Step 6

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{u}$ as $u^{\frac{1}{2}}$ .

$\frac{1}{8} \frac{1}{2} y^{2}]_{- 8}^{0} + \int_{1}^{9} u^{\frac{1}{2}} d u$

Step 7

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

$\frac{1}{8} \frac{1}{2} y^{2}]_{- 8}^{0} + \frac{2}{3} u^{\frac{3}{2}}]_{1}^{9}$

Step 8

Substitute and simplify.

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Step 8.1

Evaluate $\frac{1}{2} y^{2}$ at $0$ and at $- 8$ .

$\frac{1}{8} ((\frac{1}{2} \cdot 0^{2}) - \frac{1}{2} {(- 8)}^{2}) + \frac{2}{3} u^{\frac{3}{2}}]_{1}^{9}$

Step 8.2

Evaluate $\frac{2}{3} u^{\frac{3}{2}}$ at $9$ and at $1$ .

$\frac{1}{8} ((\frac{1}{2} \cdot 0^{2}) - \frac{1}{2} {(- 8)}^{2}) + (\frac{2}{3} \cdot 9^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3

Simplify.

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Step 8.3.1

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

$\frac{1}{8} (\frac{1}{2} \cdot 0 - \frac{1}{2} {(- 8)}^{2}) + (\frac{2}{3} \cdot 9^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.2

Multiply $\frac{1}{2}$ by $0$ .

$\frac{1}{8} (0 - \frac{1}{2} {(- 8)}^{2}) + (\frac{2}{3} \cdot 9^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.3

Raise $- 8$ to the power of $2$ .

$\frac{1}{8} (0 - \frac{1}{2} \cdot 64) + (\frac{2}{3} \cdot 9^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.4

Multiply $64$ by $- 1$ .

$\frac{1}{8} (0 - 64 (\frac{1}{2})) + (\frac{2}{3} \cdot 9^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.5

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

$\frac{1}{8} (0 + \frac{- 64}{2}) + (\frac{2}{3} \cdot 9^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.6

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

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Step 8.3.6.1

Factor $2$ out of $- 64$ .

$\frac{1}{8} (0 + \frac{2 \cdot - 32}{2}) + (\frac{2}{3} \cdot 9^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.6.2

Cancel the common factors.

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Step 8.3.6.2.1

Factor $2$ out of $2$ .

$\frac{1}{8} (0 + \frac{2 \cdot - 32}{2 (1)}) + (\frac{2}{3} \cdot 9^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.6.2.2

Cancel the common factor.

$\frac{1}{8} (0 + \frac{2 \cdot - 32}{2 \cdot 1}) + (\frac{2}{3} \cdot 9^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.6.2.3

Rewrite the expression.

$\frac{1}{8} (0 + \frac{- 32}{1}) + (\frac{2}{3} \cdot 9^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.6.2.4

Divide $- 32$ by $1$ .

$\frac{1}{8} (0 - 32) + (\frac{2}{3} \cdot 9^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.7

Subtract $32$ from $0$ .

$\frac{1}{8} \cdot - 32 + (\frac{2}{3} \cdot 9^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.8

Combine $\frac{1}{8}$ and $- 32$ .

$\frac{- 32}{8} + (\frac{2}{3} \cdot 9^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.9

Cancel the common factor of $- 32$ and $8$ .

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Step 8.3.9.1

Factor $8$ out of $- 32$ .

$\frac{8 \cdot - 4}{8} + (\frac{2}{3} \cdot 9^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.9.2

Cancel the common factors.

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Step 8.3.9.2.1

Factor $8$ out of $8$ .

$\frac{8 \cdot - 4}{8 (1)} + (\frac{2}{3} \cdot 9^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.9.2.2

Cancel the common factor.

$\frac{8 \cdot - 4}{8 \cdot 1} + (\frac{2}{3} \cdot 9^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.9.2.3

Rewrite the expression.

$\frac{- 4}{1} + (\frac{2}{3} \cdot 9^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.9.2.4

Divide $- 4$ by $1$ .

$- 4 + (\frac{2}{3} \cdot 9^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.10

Rewrite $9$ as $3^{2}$ .

$- 4 + \frac{2}{3} \cdot {(3^{2})}^{\frac{3}{2}} - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.11

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

$- 4 + \frac{2}{3} \cdot 3^{2 (\frac{3}{2})} - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.12

Cancel the common factor of $2$ .

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Step 8.3.12.1

Cancel the common factor.

$- 4 + \frac{2}{3} \cdot 3^{2 (\frac{3}{2})} - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.12.2

Rewrite the expression.

$- 4 + \frac{2}{3} \cdot 3^{3} - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.13

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

$- 4 + \frac{2}{3} \cdot 27 - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.14

Combine $\frac{2}{3}$ and $27$ .

$- 4 + \frac{2 \cdot 27}{3} - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.15

Multiply $2$ by $27$ .

$- 4 + \frac{54}{3} - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.16

Cancel the common factor of $54$ and $3$ .

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Step 8.3.16.1

Factor $3$ out of $54$ .

$- 4 + \frac{3 \cdot 18}{3} - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.16.2

Cancel the common factors.

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Step 8.3.16.2.1

Factor $3$ out of $3$ .

$- 4 + \frac{3 \cdot 18}{3 (1)} - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.16.2.2

Cancel the common factor.

$- 4 + \frac{3 \cdot 18}{3 \cdot 1} - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.16.2.3

Rewrite the expression.

$- 4 + \frac{18}{1} - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.16.2.4

Divide $18$ by $1$ .

$- 4 + 18 - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.17

One to any power is one.

$- 4 + 18 - \frac{2}{3} \cdot 1$

Step 8.3.18

Multiply $- 1$ by $1$ .

$- 4 + 18 - \frac{2}{3}$

Step 8.3.19

To write $18$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- 4 + 18 \cdot \frac{3}{3} - \frac{2}{3}$

Step 8.3.20

Combine $18$ and $\frac{3}{3}$ .

$- 4 + \frac{18 \cdot 3}{3} - \frac{2}{3}$

Step 8.3.21

Combine the numerators over the common denominator.

$- 4 + \frac{18 \cdot 3 - 2}{3}$

Step 8.3.22

Simplify the numerator.

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Step 8.3.22.1

Multiply $18$ by $3$ .

$- 4 + \frac{54 - 2}{3}$

Step 8.3.22.2

Subtract $2$ from $54$ .

$- 4 + \frac{52}{3}$

Step 8.3.23

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

$- 4 \cdot \frac{3}{3} + \frac{52}{3}$

Step 8.3.24

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

$\frac{- 4 \cdot 3}{3} + \frac{52}{3}$

Step 8.3.25

Combine the numerators over the common denominator.

$\frac{- 4 \cdot 3 + 52}{3}$

Step 8.3.26

Simplify the numerator.

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Step 8.3.26.1

Multiply $- 4$ by $3$ .

$\frac{- 12 + 52}{3}$

Step 8.3.26.2

Add $- 12$ and $52$ .

$\frac{40}{3}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{40}{3}$

Decimal Form:

$13.\overline{3}$

Mixed Number Form:

$13 \frac{1}{3}$

Step 10