Calculus Examples

$\int_{- 2}^{6} [4 y - (y^{2} - 12)] d y$

Step 1

Remove parentheses.

$\int_{- 2}^{6} 4 y - (y^{2} - 12) d y$

Step 2

Split the single integral into multiple integrals.

$\int_{- 2}^{6} 4 y d y + \int_{- 2}^{6} - (y^{2} - 12) d y$

Step 3

Since $4$ is constant with respect to $y$ , move $4$ out of the integral.

$4 \int_{- 2}^{6} y d y + \int_{- 2}^{6} - (y^{2} - 12) d y$

Step 4

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

$4 (\frac{1}{2} y^{2}]_{- 2}^{6}) + \int_{- 2}^{6} - (y^{2} - 12) d y$

Step 5

Combine $\frac{1}{2}$ and $y^{2}$ .

$4 (\frac{y^{2}}{2}]_{- 2}^{6}) + \int_{- 2}^{6} - (y^{2} - 12) d y$

Step 6

Multiply $- (y^{2} - 12)$ .

$4 (\frac{y^{2}}{2}]_{- 2}^{6}) + \int_{- 2}^{6} - y^{2} - - 12 d y$

Step 7

Multiply $- 1$ by $- 12$ .

$4 (\frac{y^{2}}{2}]_{- 2}^{6}) + \int_{- 2}^{6} - y^{2} + 12 d y$

Step 8

Split the single integral into multiple integrals.

$4 (\frac{y^{2}}{2}]_{- 2}^{6}) + \int_{- 2}^{6} - y^{2} d y + \int_{- 2}^{6} 12 d y$

Step 9

Since $- 1$ is constant with respect to $y$ , move $- 1$ out of the integral.

$4 (\frac{y^{2}}{2}]_{- 2}^{6}) - \int_{- 2}^{6} y^{2} d y + \int_{- 2}^{6} 12 d y$

Step 10

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

$4 (\frac{y^{2}}{2}]_{- 2}^{6}) - (\frac{1}{3} y^{3}]_{- 2}^{6}) + \int_{- 2}^{6} 12 d y$

Step 11

Combine $\frac{1}{3}$ and $y^{3}$ .

$4 (\frac{y^{2}}{2}]_{- 2}^{6}) - (\frac{y^{3}}{3}]_{- 2}^{6}) + \int_{- 2}^{6} 12 d y$

Step 12

Apply the constant rule.

$4 (\frac{y^{2}}{2}]_{- 2}^{6}) - (\frac{y^{3}}{3}]_{- 2}^{6}) + 12 y]_{- 2}^{6}$

Step 13

Substitute and simplify.

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Step 13.1

Evaluate $\frac{y^{2}}{2}$ at $6$ and at $- 2$ .

$4 ((\frac{6^{2}}{2}) - \frac{{(- 2)}^{2}}{2}) - (\frac{y^{3}}{3}]_{- 2}^{6}) + 12 y]_{- 2}^{6}$

Step 13.2

Evaluate $\frac{y^{3}}{3}$ at $6$ and at $- 2$ .

$4 (\frac{6^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - (\frac{6^{3}}{3} - \frac{{(- 2)}^{3}}{3}) + 12 y]_{- 2}^{6}$

Step 13.3

Evaluate $12 y$ at $6$ and at $- 2$ .

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

Step 13.4

Simplify.

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Step 13.4.1

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

$4 (\frac{36}{2} - \frac{{(- 2)}^{2}}{2}) - (\frac{6^{3}}{3} - \frac{{(- 2)}^{3}}{3}) + (12 \cdot 6) - 12 \cdot - 2$

Step 13.4.2

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

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Step 13.4.2.1

Factor $2$ out of $36$ .

$4 (\frac{2 \cdot 18}{2} - \frac{{(- 2)}^{2}}{2}) - (\frac{6^{3}}{3} - \frac{{(- 2)}^{3}}{3}) + (12 \cdot 6) - 12 \cdot - 2$

Step 13.4.2.2

Cancel the common factors.

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Step 13.4.2.2.1

Factor $2$ out of $2$ .

$4 (\frac{2 \cdot 18}{2 (1)} - \frac{{(- 2)}^{2}}{2}) - (\frac{6^{3}}{3} - \frac{{(- 2)}^{3}}{3}) + (12 \cdot 6) - 12 \cdot - 2$

Step 13.4.2.2.2

Cancel the common factor.

$4 (\frac{2 \cdot 18}{2 \cdot 1} - \frac{{(- 2)}^{2}}{2}) - (\frac{6^{3}}{3} - \frac{{(- 2)}^{3}}{3}) + (12 \cdot 6) - 12 \cdot - 2$

Step 13.4.2.2.3

Rewrite the expression.

$4 (\frac{18}{1} - \frac{{(- 2)}^{2}}{2}) - (\frac{6^{3}}{3} - \frac{{(- 2)}^{3}}{3}) + (12 \cdot 6) - 12 \cdot - 2$

Step 13.4.2.2.4

Divide $18$ by $1$ .

$4 (18 - \frac{{(- 2)}^{2}}{2}) - (\frac{6^{3}}{3} - \frac{{(- 2)}^{3}}{3}) + (12 \cdot 6) - 12 \cdot - 2$

Step 13.4.3

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

$4 (18 - \frac{4}{2}) - (\frac{6^{3}}{3} - \frac{{(- 2)}^{3}}{3}) + (12 \cdot 6) - 12 \cdot - 2$

Step 13.4.4

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

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Step 13.4.4.1

Factor $2$ out of $4$ .

$4 (18 - \frac{2 \cdot 2}{2}) - (\frac{6^{3}}{3} - \frac{{(- 2)}^{3}}{3}) + (12 \cdot 6) - 12 \cdot - 2$

Step 13.4.4.2

Cancel the common factors.

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Step 13.4.4.2.1

Factor $2$ out of $2$ .

$4 (18 - \frac{2 \cdot 2}{2 (1)}) - (\frac{6^{3}}{3} - \frac{{(- 2)}^{3}}{3}) + (12 \cdot 6) - 12 \cdot - 2$

Step 13.4.4.2.2

Cancel the common factor.

$4 (18 - \frac{2 \cdot 2}{2 \cdot 1}) - (\frac{6^{3}}{3} - \frac{{(- 2)}^{3}}{3}) + (12 \cdot 6) - 12 \cdot - 2$

Step 13.4.4.2.3

Rewrite the expression.

$4 (18 - \frac{2}{1}) - (\frac{6^{3}}{3} - \frac{{(- 2)}^{3}}{3}) + (12 \cdot 6) - 12 \cdot - 2$

Step 13.4.4.2.4

Divide $2$ by $1$ .

$4 (18 - 1 \cdot 2) - (\frac{6^{3}}{3} - \frac{{(- 2)}^{3}}{3}) + (12 \cdot 6) - 12 \cdot - 2$

Step 13.4.5

Multiply $- 1$ by $2$ .

$4 (18 - 2) - (\frac{6^{3}}{3} - \frac{{(- 2)}^{3}}{3}) + (12 \cdot 6) - 12 \cdot - 2$

Step 13.4.6

Subtract $2$ from $18$ .

$4 \cdot 16 - (\frac{6^{3}}{3} - \frac{{(- 2)}^{3}}{3}) + (12 \cdot 6) - 12 \cdot - 2$

Step 13.4.7

Multiply $4$ by $16$ .

$64 - (\frac{6^{3}}{3} - \frac{{(- 2)}^{3}}{3}) + (12 \cdot 6) - 12 \cdot - 2$

Step 13.4.8

Raise $6$ to the power of $3$ .

$64 - (\frac{216}{3} - \frac{{(- 2)}^{3}}{3}) + (12 \cdot 6) - 12 \cdot - 2$

Step 13.4.9

Cancel the common factor of $216$ and $3$ .

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Step 13.4.9.1

Factor $3$ out of $216$ .

$64 - (\frac{3 \cdot 72}{3} - \frac{{(- 2)}^{3}}{3}) + (12 \cdot 6) - 12 \cdot - 2$

Step 13.4.9.2

Cancel the common factors.

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Step 13.4.9.2.1

Factor $3$ out of $3$ .

$64 - (\frac{3 \cdot 72}{3 (1)} - \frac{{(- 2)}^{3}}{3}) + (12 \cdot 6) - 12 \cdot - 2$

Step 13.4.9.2.2

Cancel the common factor.

$64 - (\frac{3 \cdot 72}{3 \cdot 1} - \frac{{(- 2)}^{3}}{3}) + (12 \cdot 6) - 12 \cdot - 2$

Step 13.4.9.2.3

Rewrite the expression.

$64 - (\frac{72}{1} - \frac{{(- 2)}^{3}}{3}) + (12 \cdot 6) - 12 \cdot - 2$

Step 13.4.9.2.4

Divide $72$ by $1$ .

$64 - (72 - \frac{{(- 2)}^{3}}{3}) + (12 \cdot 6) - 12 \cdot - 2$

Step 13.4.10

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

$64 - (72 - \frac{- 8}{3}) + (12 \cdot 6) - 12 \cdot - 2$

Step 13.4.11

Move the negative in front of the fraction.

$64 - (72 - - \frac{8}{3}) + (12 \cdot 6) - 12 \cdot - 2$

Step 13.4.12

Multiply $- 1$ by $- 1$ .

$64 - (72 + 1 (\frac{8}{3})) + (12 \cdot 6) - 12 \cdot - 2$

Step 13.4.13

Multiply $\frac{8}{3}$ by $1$ .

$64 - (72 + \frac{8}{3}) + (12 \cdot 6) - 12 \cdot - 2$

Step 13.4.14

To write $72$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$64 - (72 \cdot \frac{3}{3} + \frac{8}{3}) + (12 \cdot 6) - 12 \cdot - 2$

Step 13.4.15

Combine $72$ and $\frac{3}{3}$ .

$64 - (\frac{72 \cdot 3}{3} + \frac{8}{3}) + (12 \cdot 6) - 12 \cdot - 2$

Step 13.4.16

Combine the numerators over the common denominator.

$64 - \frac{72 \cdot 3 + 8}{3} + (12 \cdot 6) - 12 \cdot - 2$

Step 13.4.17

Simplify the numerator.

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Step 13.4.17.1

Multiply $72$ by $3$ .

$64 - \frac{216 + 8}{3} + (12 \cdot 6) - 12 \cdot - 2$

Step 13.4.17.2

Add $216$ and $8$ .

$64 - \frac{224}{3} + (12 \cdot 6) - 12 \cdot - 2$

Step 13.4.18

To write $64$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$64 \cdot \frac{3}{3} - \frac{224}{3} + (12 \cdot 6) - 12 \cdot - 2$

Step 13.4.19

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

$\frac{64 \cdot 3}{3} - \frac{224}{3} + (12 \cdot 6) - 12 \cdot - 2$

Step 13.4.20

Combine the numerators over the common denominator.

$\frac{64 \cdot 3 - 224}{3} + (12 \cdot 6) - 12 \cdot - 2$

Step 13.4.21

Simplify the numerator.

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Step 13.4.21.1

Multiply $64$ by $3$ .

$\frac{192 - 224}{3} + (12 \cdot 6) - 12 \cdot - 2$

Step 13.4.21.2

Subtract $224$ from $192$ .

$\frac{- 32}{3} + (12 \cdot 6) - 12 \cdot - 2$

Step 13.4.22

Move the negative in front of the fraction.

$- \frac{32}{3} + (12 \cdot 6) - 12 \cdot - 2$

Step 13.4.23

Multiply $12$ by $6$ .

$- \frac{32}{3} + 72 - 12 \cdot - 2$

Step 13.4.24

Multiply $- 12$ by $- 2$ .

$- \frac{32}{3} + 72 + 24$

Step 13.4.25

Add $72$ and $24$ .

$- \frac{32}{3} + 96$

Step 13.4.26

To write $96$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{32}{3} + 96 \cdot \frac{3}{3}$

Step 13.4.27

Combine $96$ and $\frac{3}{3}$ .

$- \frac{32}{3} + \frac{96 \cdot 3}{3}$

Step 13.4.28

Combine the numerators over the common denominator.

$\frac{- 32 + 96 \cdot 3}{3}$

Step 13.4.29

Simplify the numerator.

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Step 13.4.29.1

Multiply $96$ by $3$ .

$\frac{- 32 + 288}{3}$

Step 13.4.29.2

Add $- 32$ and $288$ .

$\frac{256}{3}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$\frac{256}{3}$

Decimal Form:

$85.\overline{3}$

Mixed Number Form:

$85 \frac{1}{3}$

Step 15