Calculus Examples

$\int_{- 7}^{0} (\frac{y}{7} + \sqrt{y + 8}) d y$

Step 1

Remove parentheses.

$\int_{- 7}^{0} \frac{y}{7} + \sqrt{y + 8} d y$

Step 2

Split the single integral into multiple integrals.

$\int_{- 7}^{0} \frac{y}{7} d y + \int_{- 7}^{0} \sqrt{y + 8} d y$

Step 3

Since $\frac{1}{7}$ is constant with respect to $y$ , move $\frac{1}{7}$ out of the integral.

$\frac{1}{7} \int_{- 7}^{0} y d y + \int_{- 7}^{0} \sqrt{y + 8} d y$

Step 4

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

$\frac{1}{7} \frac{1}{2} y^{2}]_{- 7}^{0} + \int_{- 7}^{0} \sqrt{y + 8} d y$

Step 5

Let $u = y + 8$ . Then $d u = d y$ . Rewrite using $u$ and $d$ $u$ .

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Step 5.1

Let $u = y + 8$ . Find $\frac{d u}{d y}$ .

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Step 5.1.1

Differentiate $y + 8$ .

$\frac{d}{d y} [y + 8]$

Step 5.1.2

By the Sum Rule, the derivative of $y + 8$ with respect to $y$ is $\frac{d}{d y} [y] + \frac{d}{d y} [8]$ .

$\frac{d}{d y} [y] + \frac{d}{d y} [8]$

Step 5.1.3

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

$1 + \frac{d}{d y} [8]$

Step 5.1.4

Since $8$ is constant with respect to $y$ , the derivative of $8$ 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 $u = y + 8$ .

$u_{lower} = - 7 + 8$

Step 5.3

Add $- 7$ and $8$ .

$u_{lower} = 1$

Step 5.4

Substitute the upper limit in for $y$ in $u = y + 8$ .

$u_{upper} = 0 + 8$

Step 5.5

Add $0$ and $8$ .

$u_{upper} = 8$

Step 5.6

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

$u_{lower} = 1$

$u_{upper} = 8$

Step 5.7

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

$\frac{1}{7} \frac{1}{2} y^{2}]_{- 7}^{0} + \int_{1}^{8} \sqrt{u} d u$

Step 6

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

$\frac{1}{7} \frac{1}{2} y^{2}]_{- 7}^{0} + \int_{1}^{8} u^{\frac{1}{2}} d u$

Step 7

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

$\frac{1}{7} \frac{1}{2} y^{2}]_{- 7}^{0} + \frac{2}{3} u^{\frac{3}{2}}]_{1}^{8}$

Step 8

Substitute and simplify.

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Step 8.1

Evaluate $\frac{1}{2} y^{2}$ at $0$ and at $- 7$ .

$\frac{1}{7} ((\frac{1}{2} \cdot 0^{2}) - \frac{1}{2} {(- 7)}^{2}) + \frac{2}{3} u^{\frac{3}{2}}]_{1}^{8}$

Step 8.2

Evaluate $\frac{2}{3} u^{\frac{3}{2}}$ at $8$ and at $1$ .

$\frac{1}{7} ((\frac{1}{2} \cdot 0^{2}) - \frac{1}{2} {(- 7)}^{2}) + (\frac{2}{3} \cdot 8^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3

Simplify.

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Step 8.3.1

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

$\frac{1}{7} (\frac{1}{2} \cdot 0 - \frac{1}{2} {(- 7)}^{2}) + (\frac{2}{3} \cdot 8^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.2

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

$\frac{1}{7} (0 - \frac{1}{2} {(- 7)}^{2}) + (\frac{2}{3} \cdot 8^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.3

Raise $- 7$ to the power of $2$ .

$\frac{1}{7} (0 - \frac{1}{2} \cdot 49) + (\frac{2}{3} \cdot 8^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.4

Multiply $49$ by $- 1$ .

$\frac{1}{7} (0 - 49 (\frac{1}{2})) + (\frac{2}{3} \cdot 8^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.5

Combine $- 49$ and $\frac{1}{2}$ .

$\frac{1}{7} (0 + \frac{- 49}{2}) + (\frac{2}{3} \cdot 8^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.6

Move the negative in front of the fraction.

$\frac{1}{7} (0 - \frac{49}{2}) + (\frac{2}{3} \cdot 8^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.7

Subtract $\frac{49}{2}$ from $0$ .

$\frac{1}{7} (- \frac{49}{2}) + (\frac{2}{3} \cdot 8^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.8

Multiply $\frac{1}{7}$ by $\frac{49}{2}$ .

$- \frac{49}{7 \cdot 2} + (\frac{2}{3} \cdot 8^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.9

Multiply $7$ by $2$ .

$- \frac{49}{14} + (\frac{2}{3} \cdot 8^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.10

Cancel the common factor of $49$ and $14$ .

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Step 8.3.10.1

Factor $7$ out of $49$ .

$- \frac{7 (7)}{14} + (\frac{2}{3} \cdot 8^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.10.2

Cancel the common factors.

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Step 8.3.10.2.1

Factor $7$ out of $14$ .

$- \frac{7 \cdot 7}{7 \cdot 2} + (\frac{2}{3} \cdot 8^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.10.2.2

Cancel the common factor.

$- \frac{7 \cdot 7}{7 \cdot 2} + (\frac{2}{3} \cdot 8^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.10.2.3

Rewrite the expression.

$- \frac{7}{2} + (\frac{2}{3} \cdot 8^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.11

Combine $\frac{2}{3}$ and $8^{\frac{3}{2}}$ .

$- \frac{7}{2} + \frac{2 \cdot 8^{\frac{3}{2}}}{3} - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.12

Rewrite $8$ as $2^{3}$ .

$- \frac{7}{2} + \frac{2 \cdot {(2^{3})}^{\frac{3}{2}}}{3} - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.13

Multiply the exponents in ${(2^{3})}^{\frac{3}{2}}$ .

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Step 8.3.13.1

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

$- \frac{7}{2} + \frac{2 \cdot 2^{3 (\frac{3}{2})}}{3} - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.13.2

Multiply $3 (\frac{3}{2})$ .

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Step 8.3.13.2.1

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

$- \frac{7}{2} + \frac{2 \cdot 2^{\frac{3 \cdot 3}{2}}}{3} - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.13.2.2

Multiply $3$ by $3$ .

$- \frac{7}{2} + \frac{2 \cdot 2^{\frac{9}{2}}}{3} - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.14

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

$- \frac{7}{2} + \frac{2^{1 + \frac{9}{2}}}{3} - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.15

Write $1$ as a fraction with a common denominator.

$- \frac{7}{2} + \frac{2^{\frac{2}{2} + \frac{9}{2}}}{3} - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.16

Combine the numerators over the common denominator.

$- \frac{7}{2} + \frac{2^{\frac{2 + 9}{2}}}{3} - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.17

Add $2$ and $9$ .

$- \frac{7}{2} + \frac{2^{\frac{11}{2}}}{3} - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 8.3.18

One to any power is one.

$- \frac{7}{2} + \frac{2^{\frac{11}{2}}}{3} - \frac{2}{3} \cdot 1$

Step 8.3.19

Multiply $- 1$ by $1$ .

$- \frac{7}{2} + \frac{2^{\frac{11}{2}}}{3} - \frac{2}{3}$

Step 8.3.20

Combine the numerators over the common denominator.

$- \frac{7}{2} + \frac{2^{\frac{11}{2}} - 2}{3}$

Step 8.3.21

To write $- \frac{7}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{7}{2} \cdot \frac{3}{3} + \frac{2^{\frac{11}{2}} - 2}{3}$

Step 8.3.22

To write $\frac{2^{\frac{11}{2}} - 2}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{7}{2} \cdot \frac{3}{3} + \frac{2^{\frac{11}{2}} - 2}{3} \cdot \frac{2}{2}$

Step 8.3.23

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.3.23.1

Multiply $\frac{7}{2}$ by $\frac{3}{3}$ .

$- \frac{7 \cdot 3}{2 \cdot 3} + \frac{2^{\frac{11}{2}} - 2}{3} \cdot \frac{2}{2}$

Step 8.3.23.2

Multiply $2$ by $3$ .

$- \frac{7 \cdot 3}{6} + \frac{2^{\frac{11}{2}} - 2}{3} \cdot \frac{2}{2}$

Step 8.3.23.3

Multiply $\frac{2^{\frac{11}{2}} - 2}{3}$ by $\frac{2}{2}$ .

$- \frac{7 \cdot 3}{6} + \frac{(2^{\frac{11}{2}} - 2) \cdot 2}{3 \cdot 2}$

Step 8.3.23.4

Multiply $3$ by $2$ .

$- \frac{7 \cdot 3}{6} + \frac{(2^{\frac{11}{2}} - 2) \cdot 2}{6}$

Step 8.3.24

Combine the numerators over the common denominator.

$\frac{- 7 \cdot 3 + (2^{\frac{11}{2}} - 2) \cdot 2}{6}$

Step 8.3.25

Multiply $- 7$ by $3$ .

$\frac{- 21 + (2^{\frac{11}{2}} - 2) \cdot 2}{6}$

Step 8.3.26

Move $2$ to the left of $2^{\frac{11}{2}} - 2$ .

$\frac{- 21 + 2 (2^{\frac{11}{2}} - 2)}{6}$

Step 9

Simplify.

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Step 9.1

Rewrite $- 21$ as $- 1 (21)$ .

$\frac{- 1 (21) + 2 (2^{\frac{11}{2}} - 2)}{6}$

Step 9.2

Factor $- 1$ out of $2 (2^{\frac{11}{2}} - 2)$ .

$\frac{- 1 (21) - (- 2 (2^{\frac{11}{2}} - 2))}{6}$

Step 9.3

Factor $- 1$ out of $- 1 (21) - (- 2 (2^{\frac{11}{2}} - 2))$ .

$\frac{- 1 (21 - 2 (2^{\frac{11}{2}} - 2))}{6}$

Step 9.4

Move the negative in front of the fraction.

$- \frac{21 - 2 (2^{\frac{11}{2}} - 2)}{6}$

Step 10

Simplify the numerator.

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Step 10.1

Apply the distributive property.

$- \frac{21 - 2 \cdot 2^{\frac{11}{2}} - 2 \cdot - 2}{6}$

Step 10.2

Multiply $- 2 \cdot 2^{\frac{11}{2}}$ .

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Step 10.2.1

Factor out negative.

$- \frac{21 - (2 \cdot 2^{\frac{11}{2}}) - 2 \cdot - 2}{6}$

Step 10.2.2

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

$- \frac{21 - (2^{1} \cdot 2^{\frac{11}{2}}) - 2 \cdot - 2}{6}$

Step 10.2.3

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

$- \frac{21 - 2^{1 + \frac{11}{2}} - 2 \cdot - 2}{6}$

Step 10.2.4

Write $1$ as a fraction with a common denominator.

$- \frac{21 - 2^{\frac{2}{2} + \frac{11}{2}} - 2 \cdot - 2}{6}$

Step 10.2.5

Combine the numerators over the common denominator.

$- \frac{21 - 2^{\frac{2 + 11}{2}} - 2 \cdot - 2}{6}$

Step 10.2.6

Add $2$ and $11$ .

$- \frac{21 - 2^{\frac{13}{2}} - 2 \cdot - 2}{6}$

Step 10.3

Multiply $- 2$ by $- 2$ .

$- \frac{21 - 2^{\frac{13}{2}} + 4}{6}$

Step 10.4

Add $21$ and $4$ .

$- \frac{25 - 2^{\frac{13}{2}}}{6}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$- \frac{25 - 2^{\frac{13}{2}}}{6}$

Decimal Form:

$10.91827799 \dots$

Step 12