Calculus Examples

$\int_{- 2}^{4} 2 + \frac{1}{2} y - \frac{1}{4} y^{2} d y$

Step 1

Split the single integral into multiple integrals.

$\int_{- 2}^{4} 2 d y + \int_{- 2}^{4} \frac{1}{2} y d y + \int_{- 2}^{4} - \frac{1}{4} y^{2} d y$

Step 2

Apply the constant rule.

$2 y]_{- 2}^{4} + \int_{- 2}^{4} \frac{1}{2} y d y + \int_{- 2}^{4} - \frac{1}{4} y^{2} d y$

Step 3

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

$2 y]_{- 2}^{4} + \frac{1}{2} \int_{- 2}^{4} y d y + \int_{- 2}^{4} - \frac{1}{4} y^{2} d y$

Step 4

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

$2 y]_{- 2}^{4} + \frac{1}{2} \frac{1}{2} y^{2}]_{- 2}^{4} + \int_{- 2}^{4} - \frac{1}{4} y^{2} d y$

Step 5

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

$2 y]_{- 2}^{4} + \frac{1}{2} \frac{1}{2} y^{2}]_{- 2}^{4} - \frac{1}{4} \int_{- 2}^{4} y^{2} d y$

Step 6

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

$2 y]_{- 2}^{4} + \frac{1}{2} \frac{1}{2} y^{2}]_{- 2}^{4} - \frac{1}{4} \frac{1}{3} y^{3}]_{- 2}^{4}$

Step 7

Substitute and simplify.

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Step 7.1

Evaluate $2 y$ at $4$ and at $- 2$ .

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

Step 7.2

Evaluate $\frac{1}{2} y^{2}$ at $4$ and at $- 2$ .

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

Step 7.3

Evaluate $\frac{1}{3} y^{3}$ at $4$ and at $- 2$ .

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

Step 7.4

Simplify.

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Step 7.4.1

Multiply $2$ by $4$ .

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

Step 7.4.2

Multiply $- 2$ by $- 2$ .

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

Step 7.4.3

Add $8$ and $4$ .

$12 + \frac{1}{2} (\frac{1}{2} \cdot 4^{2} - \frac{1}{2} {(- 2)}^{2}) - \frac{1}{4} ((\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3})$

Step 7.4.4

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

$12 + \frac{1}{2} (\frac{1}{2} \cdot 16 - \frac{1}{2} {(- 2)}^{2}) - \frac{1}{4} ((\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3})$

Step 7.4.5

Combine $\frac{1}{2}$ and $16$ .

$12 + \frac{1}{2} (\frac{16}{2} - \frac{1}{2} {(- 2)}^{2}) - \frac{1}{4} ((\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3})$

Step 7.4.6

Cancel the common factor of $16$ and $2$ .

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Step 7.4.6.1

Factor $2$ out of $16$ .

$12 + \frac{1}{2} (\frac{2 \cdot 8}{2} - \frac{1}{2} {(- 2)}^{2}) - \frac{1}{4} ((\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3})$

Step 7.4.6.2

Cancel the common factors.

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Step 7.4.6.2.1

Factor $2$ out of $2$ .

$12 + \frac{1}{2} (\frac{2 \cdot 8}{2 (1)} - \frac{1}{2} {(- 2)}^{2}) - \frac{1}{4} ((\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3})$

Step 7.4.6.2.2

Cancel the common factor.

$12 + \frac{1}{2} (\frac{2 \cdot 8}{2 \cdot 1} - \frac{1}{2} {(- 2)}^{2}) - \frac{1}{4} ((\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3})$

Step 7.4.6.2.3

Rewrite the expression.

$12 + \frac{1}{2} (\frac{8}{1} - \frac{1}{2} {(- 2)}^{2}) - \frac{1}{4} ((\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3})$

Step 7.4.6.2.4

Divide $8$ by $1$ .

$12 + \frac{1}{2} (8 - \frac{1}{2} {(- 2)}^{2}) - \frac{1}{4} ((\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3})$

Step 7.4.7

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

$12 + \frac{1}{2} (8 - \frac{1}{2} \cdot 4) - \frac{1}{4} ((\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3})$

Step 7.4.8

Multiply $4$ by $- 1$ .

$12 + \frac{1}{2} (8 - 4 (\frac{1}{2})) - \frac{1}{4} ((\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3})$

Step 7.4.9

Combine $- 4$ and $\frac{1}{2}$ .

$12 + \frac{1}{2} (8 + \frac{- 4}{2}) - \frac{1}{4} ((\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3})$

Step 7.4.10

Cancel the common factor of $- 4$ and $2$ .

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Step 7.4.10.1

Factor $2$ out of $- 4$ .

$12 + \frac{1}{2} (8 + \frac{2 \cdot - 2}{2}) - \frac{1}{4} ((\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3})$

Step 7.4.10.2

Cancel the common factors.

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Step 7.4.10.2.1

Factor $2$ out of $2$ .

$12 + \frac{1}{2} (8 + \frac{2 \cdot - 2}{2 (1)}) - \frac{1}{4} ((\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3})$

Step 7.4.10.2.2

Cancel the common factor.

$12 + \frac{1}{2} (8 + \frac{2 \cdot - 2}{2 \cdot 1}) - \frac{1}{4} ((\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3})$

Step 7.4.10.2.3

Rewrite the expression.

$12 + \frac{1}{2} (8 + \frac{- 2}{1}) - \frac{1}{4} ((\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3})$

Step 7.4.10.2.4

Divide $- 2$ by $1$ .

$12 + \frac{1}{2} (8 - 2) - \frac{1}{4} ((\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3})$

Step 7.4.11

Subtract $2$ from $8$ .

$12 + \frac{1}{2} \cdot 6 - \frac{1}{4} ((\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3})$

Step 7.4.12

Combine $\frac{1}{2}$ and $6$ .

$12 + \frac{6}{2} - \frac{1}{4} ((\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3})$

Step 7.4.13

Cancel the common factor of $6$ and $2$ .

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Step 7.4.13.1

Factor $2$ out of $6$ .

$12 + \frac{2 \cdot 3}{2} - \frac{1}{4} ((\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3})$

Step 7.4.13.2

Cancel the common factors.

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Step 7.4.13.2.1

Factor $2$ out of $2$ .

$12 + \frac{2 \cdot 3}{2 (1)} - \frac{1}{4} ((\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3})$

Step 7.4.13.2.2

Cancel the common factor.

$12 + \frac{2 \cdot 3}{2 \cdot 1} - \frac{1}{4} ((\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3})$

Step 7.4.13.2.3

Rewrite the expression.

$12 + \frac{3}{1} - \frac{1}{4} ((\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3})$

Step 7.4.13.2.4

Divide $3$ by $1$ .

$12 + 3 - \frac{1}{4} ((\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3})$

Step 7.4.14

Add $12$ and $3$ .

$15 - \frac{1}{4} ((\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3})$

Step 7.4.15

Raise $4$ to the power of $3$ .

$15 - \frac{1}{4} (\frac{1}{3} \cdot 64 - \frac{1}{3} {(- 2)}^{3})$

Step 7.4.16

Combine $\frac{1}{3}$ and $64$ .

$15 - \frac{1}{4} (\frac{64}{3} - \frac{1}{3} {(- 2)}^{3})$

Step 7.4.17

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

$15 - \frac{1}{4} (\frac{64}{3} - \frac{1}{3} \cdot - 8)$

Step 7.4.18

Multiply $- 8$ by $- 1$ .

$15 - \frac{1}{4} (\frac{64}{3} + 8 (\frac{1}{3}))$

Step 7.4.19

Combine $8$ and $\frac{1}{3}$ .

$15 - \frac{1}{4} (\frac{64}{3} + \frac{8}{3})$

Step 7.4.20

Combine the numerators over the common denominator.

$15 - \frac{1}{4} \cdot \frac{64 + 8}{3}$

Step 7.4.21

Add $64$ and $8$ .

$15 - \frac{1}{4} \cdot \frac{72}{3}$

Step 7.4.22

Cancel the common factor of $72$ and $3$ .

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Step 7.4.22.1

Factor $3$ out of $72$ .

$15 - \frac{1}{4} \cdot \frac{3 \cdot 24}{3}$

Step 7.4.22.2

Cancel the common factors.

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Step 7.4.22.2.1

Factor $3$ out of $3$ .

$15 - \frac{1}{4} \cdot \frac{3 \cdot 24}{3 (1)}$

Step 7.4.22.2.2

Cancel the common factor.

$15 - \frac{1}{4} \cdot \frac{3 \cdot 24}{3 \cdot 1}$

Step 7.4.22.2.3

Rewrite the expression.

$15 - \frac{1}{4} \cdot \frac{24}{1}$

Step 7.4.22.2.4

Divide $24$ by $1$ .

$15 - \frac{1}{4} \cdot 24$

Step 7.4.23

Multiply $24$ by $- 1$ .

$15 - 24 (\frac{1}{4})$

Step 7.4.24

Combine $- 24$ and $\frac{1}{4}$ .

$15 + \frac{- 24}{4}$

Step 7.4.25

Cancel the common factor of $- 24$ and $4$ .

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Step 7.4.25.1

Factor $4$ out of $- 24$ .

$15 + \frac{4 \cdot - 6}{4}$

Step 7.4.25.2

Cancel the common factors.

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Step 7.4.25.2.1

Factor $4$ out of $4$ .

$15 + \frac{4 \cdot - 6}{4 (1)}$

Step 7.4.25.2.2

Cancel the common factor.

$15 + \frac{4 \cdot - 6}{4 \cdot 1}$

Step 7.4.25.2.3

Rewrite the expression.

$15 + \frac{- 6}{1}$

Step 7.4.25.2.4

Divide $- 6$ by $1$ .

$15 - 6$

Step 7.4.26

Subtract $6$ from $15$ .

$9$

Step 8