Calculus Examples

$\int_{- 1}^{4} [3 y - (y^{2} - 4)] d y$

Step 1

Remove parentheses.

$\int_{- 1}^{4} 3 y - (y^{2} - 4) d y$

Step 2

Split the single integral into multiple integrals.

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

Step 3

Since $3$ is constant with respect to $y$ , move $3$ out of the integral.

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

Step 4

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

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

Step 5

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

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

Step 6

Multiply $- (y^{2} - 4)$ .

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

Step 7

Multiply $- 1$ by $- 4$ .

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

Step 8

Split the single integral into multiple integrals.

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

Step 9

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

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

Step 10

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

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

Step 11

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

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

Step 12

Apply the constant rule.

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

Step 13

Substitute and simplify.

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Step 13.1

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

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

Step 13.2

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

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

Step 13.3

Evaluate $4 y$ at $4$ and at $- 1$ .

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

Step 13.4

Simplify.

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Step 13.4.1

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

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

Step 13.4.2

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

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Step 13.4.2.1

Factor $2$ out of $16$ .

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

Step 13.4.2.2

Cancel the common factors.

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Step 13.4.2.2.1

Factor $2$ out of $2$ .

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

Step 13.4.2.2.2

Cancel the common factor.

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

Step 13.4.2.2.3

Rewrite the expression.

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

Step 13.4.2.2.4

Divide $8$ by $1$ .

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

Step 13.4.3

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

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

Step 13.4.4

To write $8$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 13.4.5

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

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

Step 13.4.6

Combine the numerators over the common denominator.

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

Step 13.4.7

Simplify the numerator.

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Step 13.4.7.1

Multiply $8$ by $2$ .

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

Step 13.4.7.2

Subtract $1$ from $16$ .

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

Step 13.4.8

Combine $3$ and $\frac{15}{2}$ .

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

Step 13.4.9

Multiply $3$ by $15$ .

$\frac{45}{2} - (\frac{4^{3}}{3} - \frac{{(- 1)}^{3}}{3}) + (4 \cdot 4) - 4 \cdot - 1$

Step 13.4.10

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

$\frac{45}{2} - (\frac{64}{3} - \frac{{(- 1)}^{3}}{3}) + (4 \cdot 4) - 4 \cdot - 1$

Step 13.4.11

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

$\frac{45}{2} - (\frac{64}{3} - \frac{- 1}{3}) + (4 \cdot 4) - 4 \cdot - 1$

Step 13.4.12

Move the negative in front of the fraction.

$\frac{45}{2} - (\frac{64}{3} - - \frac{1}{3}) + (4 \cdot 4) - 4 \cdot - 1$

Step 13.4.13

Multiply $- 1$ by $- 1$ .

$\frac{45}{2} - (\frac{64}{3} + 1 (\frac{1}{3})) + (4 \cdot 4) - 4 \cdot - 1$

Step 13.4.14

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

$\frac{45}{2} - (\frac{64}{3} + \frac{1}{3}) + (4 \cdot 4) - 4 \cdot - 1$

Step 13.4.15

Combine the numerators over the common denominator.

$\frac{45}{2} - \frac{64 + 1}{3} + (4 \cdot 4) - 4 \cdot - 1$

Step 13.4.16

Add $64$ and $1$ .

$\frac{45}{2} - \frac{65}{3} + (4 \cdot 4) - 4 \cdot - 1$

Step 13.4.17

To write $\frac{45}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{45}{2} \cdot \frac{3}{3} - \frac{65}{3} + (4 \cdot 4) - 4 \cdot - 1$

Step 13.4.18

To write $- \frac{65}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{45}{2} \cdot \frac{3}{3} - \frac{65}{3} \cdot \frac{2}{2} + (4 \cdot 4) - 4 \cdot - 1$

Step 13.4.19

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

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Step 13.4.19.1

Multiply $\frac{45}{2}$ by $\frac{3}{3}$ .

$\frac{45 \cdot 3}{2 \cdot 3} - \frac{65}{3} \cdot \frac{2}{2} + (4 \cdot 4) - 4 \cdot - 1$

Step 13.4.19.2

Multiply $2$ by $3$ .

$\frac{45 \cdot 3}{6} - \frac{65}{3} \cdot \frac{2}{2} + (4 \cdot 4) - 4 \cdot - 1$

Step 13.4.19.3

Multiply $\frac{65}{3}$ by $\frac{2}{2}$ .

$\frac{45 \cdot 3}{6} - \frac{65 \cdot 2}{3 \cdot 2} + (4 \cdot 4) - 4 \cdot - 1$

Step 13.4.19.4

Multiply $3$ by $2$ .

$\frac{45 \cdot 3}{6} - \frac{65 \cdot 2}{6} + (4 \cdot 4) - 4 \cdot - 1$

Step 13.4.20

Combine the numerators over the common denominator.

$\frac{45 \cdot 3 - 65 \cdot 2}{6} + (4 \cdot 4) - 4 \cdot - 1$

Step 13.4.21

Simplify the numerator.

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Step 13.4.21.1

Multiply $45$ by $3$ .

$\frac{135 - 65 \cdot 2}{6} + (4 \cdot 4) - 4 \cdot - 1$

Step 13.4.21.2

Multiply $- 65$ by $2$ .

$\frac{135 - 130}{6} + (4 \cdot 4) - 4 \cdot - 1$

Step 13.4.21.3

Subtract $130$ from $135$ .

$\frac{5}{6} + (4 \cdot 4) - 4 \cdot - 1$

Step 13.4.22

Multiply $4$ by $4$ .

$\frac{5}{6} + 16 - 4 \cdot - 1$

Step 13.4.23

Multiply $- 4$ by $- 1$ .

$\frac{5}{6} + 16 + 4$

Step 13.4.24

Add $16$ and $4$ .

$\frac{5}{6} + 20$

Step 13.4.25

To write $20$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{5}{6} + 20 \cdot \frac{6}{6}$

Step 13.4.26

Combine $20$ and $\frac{6}{6}$ .

$\frac{5}{6} + \frac{20 \cdot 6}{6}$

Step 13.4.27

Combine the numerators over the common denominator.

$\frac{5 + 20 \cdot 6}{6}$

Step 13.4.28

Simplify the numerator.

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Step 13.4.28.1

Multiply $20$ by $6$ .

$\frac{5 + 120}{6}$

Step 13.4.28.2

Add $5$ and $120$ .

$\frac{125}{6}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$\frac{125}{6}$

Decimal Form:

$20.8\overline{3}$

Mixed Number Form:

$20 \frac{5}{6}$

Step 15