Calculus Examples

$\int_{- 1}^{3} [2 y - (y^{2} - 3)] d y$

Step 1

Remove parentheses.

$\int_{- 1}^{3} 2 y - (y^{2} - 3) d y$

Step 2

Split the single integral into multiple integrals.

$\int_{- 1}^{3} 2 y d y + \int_{- 1}^{3} - (y^{2} - 3) d y$

Step 3

Since $2$ is constant with respect to $y$ , move $2$ out of the integral.

$2 \int_{- 1}^{3} y d y + \int_{- 1}^{3} - (y^{2} - 3) d y$

Step 4

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

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

Step 5

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

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

Step 6

Multiply $- (y^{2} - 3)$ .

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

Step 7

Multiply $- 1$ by $- 3$ .

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

Step 8

Split the single integral into multiple integrals.

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

Step 9

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

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

Step 10

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

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

Step 11

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

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

Step 12

Apply the constant rule.

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

Step 13

Substitute and simplify.

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Step 13.1

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

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

Step 13.2

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

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

Step 13.3

Evaluate $3 y$ at $3$ and at $- 1$ .

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

Step 13.4

Simplify.

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Step 13.4.1

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

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

Step 13.4.2

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

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

Step 13.4.3

Combine the numerators over the common denominator.

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

Step 13.4.4

Subtract $1$ from $9$ .

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

Step 13.4.5

Cancel the common factor of $8$ and $2$ .

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Step 13.4.5.1

Factor $2$ out of $8$ .

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

Step 13.4.5.2

Cancel the common factors.

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Step 13.4.5.2.1

Factor $2$ out of $2$ .

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

Step 13.4.5.2.2

Cancel the common factor.

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

Step 13.4.5.2.3

Rewrite the expression.

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

Step 13.4.5.2.4

Divide $4$ by $1$ .

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

Step 13.4.6

Multiply $2$ by $4$ .

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

Step 13.4.7

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

$8 - (\frac{27}{3} - \frac{{(- 1)}^{3}}{3}) + (3 \cdot 3) - 3 \cdot - 1$

Step 13.4.8

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

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Step 13.4.8.1

Factor $3$ out of $27$ .

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

Step 13.4.8.2

Cancel the common factors.

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Step 13.4.8.2.1

Factor $3$ out of $3$ .

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

Step 13.4.8.2.2

Cancel the common factor.

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

Step 13.4.8.2.3

Rewrite the expression.

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

Step 13.4.8.2.4

Divide $9$ by $1$ .

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

Step 13.4.9

Raise $- 1$ to the power of $3$ .

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

Step 13.4.10

Move the negative in front of the fraction.

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

Step 13.4.11

Multiply $- 1$ by $- 1$ .

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

Step 13.4.12

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

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

Step 13.4.13

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

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

Step 13.4.14

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

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

Step 13.4.15

Combine the numerators over the common denominator.

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

Step 13.4.16

Simplify the numerator.

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Step 13.4.16.1

Multiply $9$ by $3$ .

$8 - \frac{27 + 1}{3} + (3 \cdot 3) - 3 \cdot - 1$

Step 13.4.16.2

Add $27$ and $1$ .

$8 - \frac{28}{3} + (3 \cdot 3) - 3 \cdot - 1$

Step 13.4.17

To write $8$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$8 \cdot \frac{3}{3} - \frac{28}{3} + (3 \cdot 3) - 3 \cdot - 1$

Step 13.4.18

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

$\frac{8 \cdot 3}{3} - \frac{28}{3} + (3 \cdot 3) - 3 \cdot - 1$

Step 13.4.19

Combine the numerators over the common denominator.

$\frac{8 \cdot 3 - 28}{3} + (3 \cdot 3) - 3 \cdot - 1$

Step 13.4.20

Simplify the numerator.

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Step 13.4.20.1

Multiply $8$ by $3$ .

$\frac{24 - 28}{3} + (3 \cdot 3) - 3 \cdot - 1$

Step 13.4.20.2

Subtract $28$ from $24$ .

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

Step 13.4.21

Move the negative in front of the fraction.

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

Step 13.4.22

Multiply $3$ by $3$ .

$- \frac{4}{3} + 9 - 3 \cdot - 1$

Step 13.4.23

Multiply $- 3$ by $- 1$ .

$- \frac{4}{3} + 9 + 3$

Step 13.4.24

Add $9$ and $3$ .

$- \frac{4}{3} + 12$

Step 13.4.25

To write $12$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{4}{3} + 12 \cdot \frac{3}{3}$

Step 13.4.26

Combine $12$ and $\frac{3}{3}$ .

$- \frac{4}{3} + \frac{12 \cdot 3}{3}$

Step 13.4.27

Combine the numerators over the common denominator.

$\frac{- 4 + 12 \cdot 3}{3}$

Step 13.4.28

Simplify the numerator.

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Step 13.4.28.1

Multiply $12$ by $3$ .

$\frac{- 4 + 36}{3}$

Step 13.4.28.2

Add $- 4$ and $36$ .

$\frac{32}{3}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$\frac{32}{3}$

Decimal Form:

$10.\overline{6}$

Mixed Number Form:

$10 \frac{2}{3}$

Step 15