Calculus Examples

$\int_{- 3}^{4} [y - (y^{2} - 12)] d y$

Step 1

Remove parentheses.

$\int_{- 3}^{4} y - (y^{2} - 12) d y$

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

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

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

Step 5

Multiply $- 1$ by $- 12$ .

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

Step 6

Split the single integral into multiple integrals.

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

Apply the constant rule.

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

Step 11

Simplify the answer.

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Step 11.1

Combine $\frac{1}{2} y^{2}]_{- 3}^{4}$ and $12 y]_{- 3}^{4}$ .

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

Step 11.2

Substitute and simplify.

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Step 11.2.1

Evaluate $\frac{1}{2} y^{2} + 12 y$ at $4$ and at $- 3$ .

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

Step 11.2.2

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

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

Step 11.2.3

Simplify.

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Step 11.2.3.1

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

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

Step 11.2.3.2

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

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

Step 11.2.3.3

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

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Step 11.2.3.3.1

Factor $2$ out of $16$ .

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

Step 11.2.3.3.2

Cancel the common factors.

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Step 11.2.3.3.2.1

Factor $2$ out of $2$ .

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

Step 11.2.3.3.2.2

Cancel the common factor.

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

Step 11.2.3.3.2.3

Rewrite the expression.

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

Step 11.2.3.3.2.4

Divide $8$ by $1$ .

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

Step 11.2.3.4

Multiply $12$ by $4$ .

$8 + 48 - (\frac{1}{2} {(- 3)}^{2} + 12 \cdot - 3) - ((\frac{4^{3}}{3}) - \frac{{(- 3)}^{3}}{3})$

Step 11.2.3.5

Add $8$ and $48$ .

$56 - (\frac{1}{2} {(- 3)}^{2} + 12 \cdot - 3) - ((\frac{4^{3}}{3}) - \frac{{(- 3)}^{3}}{3})$

Step 11.2.3.6

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

$56 - (\frac{1}{2} \cdot 9 + 12 \cdot - 3) - ((\frac{4^{3}}{3}) - \frac{{(- 3)}^{3}}{3})$

Step 11.2.3.7

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

$56 - (\frac{9}{2} + 12 \cdot - 3) - ((\frac{4^{3}}{3}) - \frac{{(- 3)}^{3}}{3})$

Step 11.2.3.8

Multiply $12$ by $- 3$ .

$56 - (\frac{9}{2} - 36) - ((\frac{4^{3}}{3}) - \frac{{(- 3)}^{3}}{3})$

Step 11.2.3.9

To write $- 36$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$56 - (\frac{9}{2} - 36 \cdot \frac{2}{2}) - ((\frac{4^{3}}{3}) - \frac{{(- 3)}^{3}}{3})$

Step 11.2.3.10

Combine $- 36$ and $\frac{2}{2}$ .

$56 - (\frac{9}{2} + \frac{- 36 \cdot 2}{2}) - ((\frac{4^{3}}{3}) - \frac{{(- 3)}^{3}}{3})$

Step 11.2.3.11

Combine the numerators over the common denominator.

$56 - \frac{9 - 36 \cdot 2}{2} - ((\frac{4^{3}}{3}) - \frac{{(- 3)}^{3}}{3})$

Step 11.2.3.12

Simplify the numerator.

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Step 11.2.3.12.1

Multiply $- 36$ by $2$ .

$56 - \frac{9 - 72}{2} - ((\frac{4^{3}}{3}) - \frac{{(- 3)}^{3}}{3})$

Step 11.2.3.12.2

Subtract $72$ from $9$ .

$56 - \frac{- 63}{2} - ((\frac{4^{3}}{3}) - \frac{{(- 3)}^{3}}{3})$

Step 11.2.3.13

Move the negative in front of the fraction.

$56 - - \frac{63}{2} - ((\frac{4^{3}}{3}) - \frac{{(- 3)}^{3}}{3})$

Step 11.2.3.14

Multiply $- 1$ by $- 1$ .

$56 + 1 (\frac{63}{2}) - ((\frac{4^{3}}{3}) - \frac{{(- 3)}^{3}}{3})$

Step 11.2.3.15

Multiply $\frac{63}{2}$ by $1$ .

$56 + \frac{63}{2} - ((\frac{4^{3}}{3}) - \frac{{(- 3)}^{3}}{3})$

Step 11.2.3.16

To write $56$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$56 \cdot \frac{2}{2} + \frac{63}{2} - ((\frac{4^{3}}{3}) - \frac{{(- 3)}^{3}}{3})$

Step 11.2.3.17

Combine $56$ and $\frac{2}{2}$ .

$\frac{56 \cdot 2}{2} + \frac{63}{2} - ((\frac{4^{3}}{3}) - \frac{{(- 3)}^{3}}{3})$

Step 11.2.3.18

Combine the numerators over the common denominator.

$\frac{56 \cdot 2 + 63}{2} - ((\frac{4^{3}}{3}) - \frac{{(- 3)}^{3}}{3})$

Step 11.2.3.19

Simplify the numerator.

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Step 11.2.3.19.1

Multiply $56$ by $2$ .

$\frac{112 + 63}{2} - ((\frac{4^{3}}{3}) - \frac{{(- 3)}^{3}}{3})$

Step 11.2.3.19.2

Add $112$ and $63$ .

$\frac{175}{2} - ((\frac{4^{3}}{3}) - \frac{{(- 3)}^{3}}{3})$

Step 11.2.3.20

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

$\frac{175}{2} - (\frac{64}{3} - \frac{{(- 3)}^{3}}{3})$

Step 11.2.3.21

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

$\frac{175}{2} - (\frac{64}{3} - \frac{- 27}{3})$

Step 11.2.3.22

Cancel the common factor of $- 27$ and $3$ .

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Step 11.2.3.22.1

Factor $3$ out of $- 27$ .

$\frac{175}{2} - (\frac{64}{3} - \frac{3 \cdot - 9}{3})$

Step 11.2.3.22.2

Cancel the common factors.

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Step 11.2.3.22.2.1

Factor $3$ out of $3$ .

$\frac{175}{2} - (\frac{64}{3} - \frac{3 \cdot - 9}{3 (1)})$

Step 11.2.3.22.2.2

Cancel the common factor.

$\frac{175}{2} - (\frac{64}{3} - \frac{3 \cdot - 9}{3 \cdot 1})$

Step 11.2.3.22.2.3

Rewrite the expression.

$\frac{175}{2} - (\frac{64}{3} - \frac{- 9}{1})$

Step 11.2.3.22.2.4

Divide $- 9$ by $1$ .

$\frac{175}{2} - (\frac{64}{3} - - 9)$

Step 11.2.3.23

Multiply $- 1$ by $- 9$ .

$\frac{175}{2} - (\frac{64}{3} + 9)$

Step 11.2.3.24

To write $9$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{175}{2} - (\frac{64}{3} + 9 \cdot \frac{3}{3})$

Step 11.2.3.25

Combine $9$ and $\frac{3}{3}$ .

$\frac{175}{2} - (\frac{64}{3} + \frac{9 \cdot 3}{3})$

Step 11.2.3.26

Combine the numerators over the common denominator.

$\frac{175}{2} - \frac{64 + 9 \cdot 3}{3}$

Step 11.2.3.27

Simplify the numerator.

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Step 11.2.3.27.1

Multiply $9$ by $3$ .

$\frac{175}{2} - \frac{64 + 27}{3}$

Step 11.2.3.27.2

Add $64$ and $27$ .

$\frac{175}{2} - \frac{91}{3}$

Step 11.2.3.28

To write $\frac{175}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{175}{2} \cdot \frac{3}{3} - \frac{91}{3}$

Step 11.2.3.29

To write $- \frac{91}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{175}{2} \cdot \frac{3}{3} - \frac{91}{3} \cdot \frac{2}{2}$

Step 11.2.3.30

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

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Step 11.2.3.30.1

Multiply $\frac{175}{2}$ by $\frac{3}{3}$ .

$\frac{175 \cdot 3}{2 \cdot 3} - \frac{91}{3} \cdot \frac{2}{2}$

Step 11.2.3.30.2

Multiply $2$ by $3$ .

$\frac{175 \cdot 3}{6} - \frac{91}{3} \cdot \frac{2}{2}$

Step 11.2.3.30.3

Multiply $\frac{91}{3}$ by $\frac{2}{2}$ .

$\frac{175 \cdot 3}{6} - \frac{91 \cdot 2}{3 \cdot 2}$

Step 11.2.3.30.4

Multiply $3$ by $2$ .

$\frac{175 \cdot 3}{6} - \frac{91 \cdot 2}{6}$

Step 11.2.3.31

Combine the numerators over the common denominator.

$\frac{175 \cdot 3 - 91 \cdot 2}{6}$

Step 11.2.3.32

Simplify the numerator.

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Step 11.2.3.32.1

Multiply $175$ by $3$ .

$\frac{525 - 91 \cdot 2}{6}$

Step 11.2.3.32.2

Multiply $- 91$ by $2$ .

$\frac{525 - 182}{6}$

Step 11.2.3.32.3

Subtract $182$ from $525$ .

$\frac{343}{6}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{343}{6}$

Decimal Form:

$57.1\overline{6}$

Mixed Number Form:

$57 \frac{1}{6}$

Step 13