Calculus Examples

$\int_{0}^{3} 6 + 6 y - y^{2} d y$

Step 1

Split the single integral into multiple integrals.

$\int_{0}^{3} 6 d y + \int_{0}^{3} 6 y d y + \int_{0}^{3} - y^{2} d y$

Step 2

Apply the constant rule.

$6 y]_{0}^{3} + \int_{0}^{3} 6 y d y + \int_{0}^{3} - y^{2} d y$

Step 3

Since $6$ is constant with respect to $y$ , move $6$ out of the integral.

$6 y]_{0}^{3} + 6 \int_{0}^{3} y d y + \int_{0}^{3} - y^{2} d y$

Step 4

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

$6 y]_{0}^{3} + 6 (\frac{1}{2} y^{2}]_{0}^{3}) + \int_{0}^{3} - y^{2} d y$

Step 5

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

$6 y]_{0}^{3} + 6 (\frac{y^{2}}{2}]_{0}^{3}) + \int_{0}^{3} - y^{2} d y$

Step 6

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

$6 y]_{0}^{3} + 6 (\frac{y^{2}}{2}]_{0}^{3}) - \int_{0}^{3} y^{2} d y$

Step 7

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

$6 y]_{0}^{3} + 6 (\frac{y^{2}}{2}]_{0}^{3}) - (\frac{1}{3} y^{3}]_{0}^{3})$

Step 8

Simplify the answer.

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Step 8.1

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

$6 y]_{0}^{3} + 6 (\frac{y^{2}}{2}]_{0}^{3}) - (\frac{y^{3}}{3}]_{0}^{3})$

Step 8.2

Substitute and simplify.

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Step 8.2.1

Evaluate $6 y$ at $3$ and at $0$ .

$(6 \cdot 3) - 6 \cdot 0 + 6 (\frac{y^{2}}{2}]_{0}^{3}) - (\frac{y^{3}}{3}]_{0}^{3})$

Step 8.2.2

Evaluate $\frac{y^{2}}{2}$ at $3$ and at $0$ .

$6 \cdot 3 - 6 \cdot 0 + 6 (\frac{3^{2}}{2} - \frac{0^{2}}{2}) - (\frac{y^{3}}{3}]_{0}^{3})$

Step 8.2.3

Evaluate $\frac{y^{3}}{3}$ at $3$ and at $0$ .

$6 \cdot 3 - 6 \cdot 0 + 6 (\frac{3^{2}}{2} - \frac{0^{2}}{2}) - ((\frac{3^{3}}{3}) - \frac{0^{3}}{3})$

Step 8.2.4

Simplify.

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Step 8.2.4.1

Multiply $6$ by $3$ .

$18 - 6 \cdot 0 + 6 (\frac{3^{2}}{2} - \frac{0^{2}}{2}) - ((\frac{3^{3}}{3}) - \frac{0^{3}}{3})$

Step 8.2.4.2

Multiply $- 6$ by $0$ .

$18 + 0 + 6 (\frac{3^{2}}{2} - \frac{0^{2}}{2}) - ((\frac{3^{3}}{3}) - \frac{0^{3}}{3})$

Step 8.2.4.3

Add $18$ and $0$ .

$18 + 6 (\frac{3^{2}}{2} - \frac{0^{2}}{2}) - ((\frac{3^{3}}{3}) - \frac{0^{3}}{3})$

Step 8.2.4.4

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

$18 + 6 (\frac{9}{2} - \frac{0^{2}}{2}) - ((\frac{3^{3}}{3}) - \frac{0^{3}}{3})$

Step 8.2.4.5

Raising $0$ to any positive power yields $0$ .

$18 + 6 (\frac{9}{2} - \frac{0}{2}) - ((\frac{3^{3}}{3}) - \frac{0^{3}}{3})$

Step 8.2.4.6

Cancel the common factor of $0$ and $2$ .

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Step 8.2.4.6.1

Factor $2$ out of $0$ .

$18 + 6 (\frac{9}{2} - \frac{2 (0)}{2}) - ((\frac{3^{3}}{3}) - \frac{0^{3}}{3})$

Step 8.2.4.6.2

Cancel the common factors.

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Step 8.2.4.6.2.1

Factor $2$ out of $2$ .

$18 + 6 (\frac{9}{2} - \frac{2 \cdot 0}{2 \cdot 1}) - ((\frac{3^{3}}{3}) - \frac{0^{3}}{3})$

Step 8.2.4.6.2.2

Cancel the common factor.

$18 + 6 (\frac{9}{2} - \frac{2 \cdot 0}{2 \cdot 1}) - ((\frac{3^{3}}{3}) - \frac{0^{3}}{3})$

Step 8.2.4.6.2.3

Rewrite the expression.

$18 + 6 (\frac{9}{2} - \frac{0}{1}) - ((\frac{3^{3}}{3}) - \frac{0^{3}}{3})$

Step 8.2.4.6.2.4

Divide $0$ by $1$ .

$18 + 6 (\frac{9}{2} - 0) - ((\frac{3^{3}}{3}) - \frac{0^{3}}{3})$

Step 8.2.4.7

Multiply $- 1$ by $0$ .

$18 + 6 (\frac{9}{2} + 0) - ((\frac{3^{3}}{3}) - \frac{0^{3}}{3})$

Step 8.2.4.8

Add $\frac{9}{2}$ and $0$ .

$18 + 6 (\frac{9}{2}) - ((\frac{3^{3}}{3}) - \frac{0^{3}}{3})$

Step 8.2.4.9

Combine $6$ and $\frac{9}{2}$ .

$18 + \frac{6 \cdot 9}{2} - ((\frac{3^{3}}{3}) - \frac{0^{3}}{3})$

Step 8.2.4.10

Multiply $6$ by $9$ .

$18 + \frac{54}{2} - ((\frac{3^{3}}{3}) - \frac{0^{3}}{3})$

Step 8.2.4.11

Cancel the common factor of $54$ and $2$ .

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Step 8.2.4.11.1

Factor $2$ out of $54$ .

$18 + \frac{2 \cdot 27}{2} - ((\frac{3^{3}}{3}) - \frac{0^{3}}{3})$

Step 8.2.4.11.2

Cancel the common factors.

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Step 8.2.4.11.2.1

Factor $2$ out of $2$ .

$18 + \frac{2 \cdot 27}{2 (1)} - ((\frac{3^{3}}{3}) - \frac{0^{3}}{3})$

Step 8.2.4.11.2.2

Cancel the common factor.

$18 + \frac{2 \cdot 27}{2 \cdot 1} - ((\frac{3^{3}}{3}) - \frac{0^{3}}{3})$

Step 8.2.4.11.2.3

Rewrite the expression.

$18 + \frac{27}{1} - ((\frac{3^{3}}{3}) - \frac{0^{3}}{3})$

Step 8.2.4.11.2.4

Divide $27$ by $1$ .

$18 + 27 - ((\frac{3^{3}}{3}) - \frac{0^{3}}{3})$

Step 8.2.4.12

Add $18$ and $27$ .

$45 - ((\frac{3^{3}}{3}) - \frac{0^{3}}{3})$

Step 8.2.4.13

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

$45 - (\frac{27}{3} - \frac{0^{3}}{3})$

Step 8.2.4.14

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

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Step 8.2.4.14.1

Factor $3$ out of $27$ .

$45 - (\frac{3 \cdot 9}{3} - \frac{0^{3}}{3})$

Step 8.2.4.14.2

Cancel the common factors.

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Step 8.2.4.14.2.1

Factor $3$ out of $3$ .

$45 - (\frac{3 \cdot 9}{3 (1)} - \frac{0^{3}}{3})$

Step 8.2.4.14.2.2

Cancel the common factor.

$45 - (\frac{3 \cdot 9}{3 \cdot 1} - \frac{0^{3}}{3})$

Step 8.2.4.14.2.3

Rewrite the expression.

$45 - (\frac{9}{1} - \frac{0^{3}}{3})$

Step 8.2.4.14.2.4

Divide $9$ by $1$ .

$45 - (9 - \frac{0^{3}}{3})$

Step 8.2.4.15

Raising $0$ to any positive power yields $0$ .

$45 - (9 - \frac{0}{3})$

Step 8.2.4.16

Cancel the common factor of $0$ and $3$ .

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Step 8.2.4.16.1

Factor $3$ out of $0$ .

$45 - (9 - \frac{3 (0)}{3})$

Step 8.2.4.16.2

Cancel the common factors.

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Step 8.2.4.16.2.1

Factor $3$ out of $3$ .

$45 - (9 - \frac{3 \cdot 0}{3 \cdot 1})$

Step 8.2.4.16.2.2

Cancel the common factor.

$45 - (9 - \frac{3 \cdot 0}{3 \cdot 1})$

Step 8.2.4.16.2.3

Rewrite the expression.

$45 - (9 - \frac{0}{1})$

Step 8.2.4.16.2.4

Divide $0$ by $1$ .

$45 - (9 - 0)$

Step 8.2.4.17

Multiply $- 1$ by $0$ .

$45 - (9 + 0)$

Step 8.2.4.18

Add $9$ and $0$ .

$45 - 1 \cdot 9$

Step 8.2.4.19

Multiply $- 1$ by $9$ .

$45 - 9$

Step 8.2.4.20

Subtract $9$ from $45$ .

$36$

Step 9