Calculus Examples

$p \int_{0}^{\sqrt{6}} {(6 - y^{2})}^{2} d y$

Step 1

Expand ${(6 - y^{2})}^{2}$ .

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Step 1.1

Rewrite ${(6 - y^{2})}^{2}$ as $(6 - y^{2}) (6 - y^{2})$ .

$p \int_{0}^{\sqrt{6}} (6 - y^{2}) (6 - y^{2}) d y$

Step 1.2

Apply the distributive property.

$p \int_{0}^{\sqrt{6}} 6 (6 - y^{2}) - y^{2} (6 - y^{2}) d y$

Step 1.3

Apply the distributive property.

$p \int_{0}^{\sqrt{6}} 6 \cdot 6 + 6 (- y^{2}) - y^{2} (6 - y^{2}) d y$

Step 1.4

Apply the distributive property.

$p \int_{0}^{\sqrt{6}} 6 \cdot 6 + 6 (- y^{2}) - y^{2} \cdot 6 - y^{2} (- y^{2}) d y$

Step 1.5

Move $y^{2}$ .

$p \int_{0}^{\sqrt{6}} 6 \cdot 6 + 6 \cdot - 1 y^{2} - 1 \cdot 6 y^{2} - y^{2} (- y^{2}) d y$

Step 1.6

Move $y^{2}$ .

$p \int_{0}^{\sqrt{6}} 6 \cdot 6 + 6 \cdot - 1 y^{2} - 1 \cdot 6 y^{2} - 1 \cdot - 1 y^{2} y^{2} d y$

Step 1.7

Multiply $6$ by $6$ .

$p \int_{0}^{\sqrt{6}} 36 + 6 \cdot - 1 y^{2} - 1 \cdot 6 y^{2} - 1 \cdot - 1 y^{2} y^{2} d y$

Step 1.8

Multiply $6$ by $- 1$ .

$p \int_{0}^{\sqrt{6}} 36 - 6 y^{2} - 1 \cdot 6 y^{2} - 1 \cdot - 1 y^{2} y^{2} d y$

Step 1.9

Multiply $- 1$ by $6$ .

$p \int_{0}^{\sqrt{6}} 36 - 6 y^{2} - 6 y^{2} - 1 \cdot - 1 y^{2} y^{2} d y$

Step 1.10

Multiply $- 1$ by $- 1$ .

$p \int_{0}^{\sqrt{6}} 36 - 6 y^{2} - 6 y^{2} + 1 y^{2} y^{2} d y$

Step 1.11

Multiply $y^{2}$ by $1$ .

$p \int_{0}^{\sqrt{6}} 36 - 6 y^{2} - 6 y^{2} + y^{2} y^{2} d y$

Step 1.12

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$p \int_{0}^{\sqrt{6}} 36 - 6 y^{2} - 6 y^{2} + y^{2 + 2} d y$

Step 1.13

Add $2$ and $2$ .

$p \int_{0}^{\sqrt{6}} 36 - 6 y^{2} - 6 y^{2} + y^{4} d y$

Step 1.14

Subtract $6 y^{2}$ from $- 6 y^{2}$ .

$p \int_{0}^{\sqrt{6}} 36 - 12 y^{2} + y^{4} d y$

Step 1.15

Reorder $- 12 y^{2}$ and $y^{4}$ .

$p \int_{0}^{\sqrt{6}} 36 + y^{4} - 12 y^{2} d y$

Step 1.16

Move $36$ .

$p \int_{0}^{\sqrt{6}} y^{4} - 12 y^{2} + 36 d y$

Step 2

Split the single integral into multiple integrals.

$p (\int_{0}^{\sqrt{6}} y^{4} d y + \int_{0}^{\sqrt{6}} - 12 y^{2} d y + \int_{0}^{\sqrt{6}} 36 d y)$

Step 3

By the Power Rule, the integral of $y^{4}$ with respect to $y$ is $\frac{1}{5} y^{5}$ .

$p (\frac{1}{5} y^{5}]_{0}^{\sqrt{6}} + \int_{0}^{\sqrt{6}} - 12 y^{2} d y + \int_{0}^{\sqrt{6}} 36 d y)$

Step 4

Combine $\frac{1}{5}$ and $y^{5}$ .

$p (\frac{y^{5}}{5}]_{0}^{\sqrt{6}} + \int_{0}^{\sqrt{6}} - 12 y^{2} d y + \int_{0}^{\sqrt{6}} 36 d y)$

Step 5

Since $- 12$ is constant with respect to $y$ , move $- 12$ out of the integral.

$p (\frac{y^{5}}{5}]_{0}^{\sqrt{6}} - 12 \int_{0}^{\sqrt{6}} y^{2} d y + \int_{0}^{\sqrt{6}} 36 d y)$

Step 6

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

$p (\frac{y^{5}}{5}]_{0}^{\sqrt{6}} - 12 (\frac{1}{3} y^{3}]_{0}^{\sqrt{6}}) + \int_{0}^{\sqrt{6}} 36 d y)$

Step 7

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

$p (\frac{y^{5}}{5}]_{0}^{\sqrt{6}} - 12 (\frac{y^{3}}{3}]_{0}^{\sqrt{6}}) + \int_{0}^{\sqrt{6}} 36 d y)$

Step 8

Apply the constant rule.

$p (\frac{y^{5}}{5}]_{0}^{\sqrt{6}} - 12 (\frac{y^{3}}{3}]_{0}^{\sqrt{6}}) + 36 y]_{0}^{\sqrt{6}})$

Step 9

Simplify the answer.

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Step 9.1

Combine $\frac{y^{5}}{5}]_{0}^{\sqrt{6}}$ and $36 y]_{0}^{\sqrt{6}}$ .

$p (\frac{y^{5}}{5} + 36 y]_{0}^{\sqrt{6}} - 12 (\frac{y^{3}}{3}]_{0}^{\sqrt{6}}))$

Step 9.2

Substitute and simplify.

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Step 9.2.1

Evaluate $\frac{y^{5}}{5} + 36 y$ at $\sqrt{6}$ and at $0$ .

$p ((\frac{{\sqrt{6}}^{5}}{5} + 36 \sqrt{6}) - (\frac{0^{5}}{5} + 36 \cdot 0) - 12 (\frac{y^{3}}{3}]_{0}^{\sqrt{6}}))$

Step 9.2.2

Evaluate $\frac{y^{3}}{3}$ at $\sqrt{6}$ and at $0$ .

$p ((\frac{{\sqrt{6}}^{5}}{5} + 36 \sqrt{6}) - (\frac{0^{5}}{5} + 36 \cdot 0) - 12 ((\frac{{\sqrt{6}}^{3}}{3}) - \frac{0^{3}}{3}))$

Step 9.2.3

Simplify.

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Step 9.2.3.1

Rewrite ${\sqrt{6}}^{5}$ as $\sqrt{6^{5}}$ .

$p (\frac{\sqrt{6^{5}}}{5} + 36 \sqrt{6} - (\frac{0^{5}}{5} + 36 \cdot 0) - 12 ((\frac{{\sqrt{6}}^{3}}{3}) - \frac{0^{3}}{3}))$

Step 9.2.3.2

Raise $6$ to the power of $5$ .

$p (\frac{\sqrt{7776}}{5} + 36 \sqrt{6} - (\frac{0^{5}}{5} + 36 \cdot 0) - 12 ((\frac{{\sqrt{6}}^{3}}{3}) - \frac{0^{3}}{3}))$

Step 9.2.3.3

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

$p (\frac{\sqrt{7776}}{5} + 36 \sqrt{6} - (\frac{0}{5} + 36 \cdot 0) - 12 ((\frac{{\sqrt{6}}^{3}}{3}) - \frac{0^{3}}{3}))$

Step 9.2.3.4

Cancel the common factor of $0$ and $5$ .

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Step 9.2.3.4.1

Factor $5$ out of $0$ .

$p (\frac{\sqrt{7776}}{5} + 36 \sqrt{6} - (\frac{5 (0)}{5} + 36 \cdot 0) - 12 ((\frac{{\sqrt{6}}^{3}}{3}) - \frac{0^{3}}{3}))$

Step 9.2.3.4.2

Cancel the common factors.

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Step 9.2.3.4.2.1

Factor $5$ out of $5$ .

$p (\frac{\sqrt{7776}}{5} + 36 \sqrt{6} - (\frac{5 \cdot 0}{5 \cdot 1} + 36 \cdot 0) - 12 ((\frac{{\sqrt{6}}^{3}}{3}) - \frac{0^{3}}{3}))$

Step 9.2.3.4.2.2

Cancel the common factor.

$p (\frac{\sqrt{7776}}{5} + 36 \sqrt{6} - (\frac{5 \cdot 0}{5 \cdot 1} + 36 \cdot 0) - 12 ((\frac{{\sqrt{6}}^{3}}{3}) - \frac{0^{3}}{3}))$

Step 9.2.3.4.2.3

Rewrite the expression.

$p (\frac{\sqrt{7776}}{5} + 36 \sqrt{6} - (\frac{0}{1} + 36 \cdot 0) - 12 ((\frac{{\sqrt{6}}^{3}}{3}) - \frac{0^{3}}{3}))$

Step 9.2.3.4.2.4

Divide $0$ by $1$ .

$p (\frac{\sqrt{7776}}{5} + 36 \sqrt{6} - (0 + 36 \cdot 0) - 12 ((\frac{{\sqrt{6}}^{3}}{3}) - \frac{0^{3}}{3}))$

Step 9.2.3.5

Multiply $36$ by $0$ .

$p (\frac{\sqrt{7776}}{5} + 36 \sqrt{6} - (0 + 0) - 12 ((\frac{{\sqrt{6}}^{3}}{3}) - \frac{0^{3}}{3}))$

Step 9.2.3.6

Add $0$ and $0$ .

$p (\frac{\sqrt{7776}}{5} + 36 \sqrt{6} - 0 - 12 ((\frac{{\sqrt{6}}^{3}}{3}) - \frac{0^{3}}{3}))$

Step 9.2.3.7

Multiply $- 1$ by $0$ .

$p (\frac{\sqrt{7776}}{5} + 36 \sqrt{6} + 0 - 12 ((\frac{{\sqrt{6}}^{3}}{3}) - \frac{0^{3}}{3}))$

Step 9.2.3.8

Add $\frac{\sqrt{7776}}{5} + 36 \sqrt{6}$ and $0$ .

$p (\frac{\sqrt{7776}}{5} + 36 \sqrt{6} - 12 ((\frac{{\sqrt{6}}^{3}}{3}) - \frac{0^{3}}{3}))$

Step 9.2.3.9

Rewrite ${\sqrt{6}}^{3}$ as $\sqrt{6^{3}}$ .

$p (\frac{\sqrt{7776}}{5} + 36 \sqrt{6} - 12 (\frac{\sqrt{6^{3}}}{3} - \frac{0^{3}}{3}))$

Step 9.2.3.10

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

$p (\frac{\sqrt{7776}}{5} + 36 \sqrt{6} - 12 (\frac{\sqrt{216}}{3} - \frac{0^{3}}{3}))$

Step 9.2.3.11

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

$p (\frac{\sqrt{7776}}{5} + 36 \sqrt{6} - 12 (\frac{\sqrt{216}}{3} - \frac{0}{3}))$

Step 9.2.3.12

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

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Step 9.2.3.12.1

Factor $3$ out of $0$ .

$p (\frac{\sqrt{7776}}{5} + 36 \sqrt{6} - 12 (\frac{\sqrt{216}}{3} - \frac{3 (0)}{3}))$

Step 9.2.3.12.2

Cancel the common factors.

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Step 9.2.3.12.2.1

Factor $3$ out of $3$ .

$p (\frac{\sqrt{7776}}{5} + 36 \sqrt{6} - 12 (\frac{\sqrt{216}}{3} - \frac{3 \cdot 0}{3 \cdot 1}))$

Step 9.2.3.12.2.2

Cancel the common factor.

$p (\frac{\sqrt{7776}}{5} + 36 \sqrt{6} - 12 (\frac{\sqrt{216}}{3} - \frac{3 \cdot 0}{3 \cdot 1}))$

Step 9.2.3.12.2.3

Rewrite the expression.

$p (\frac{\sqrt{7776}}{5} + 36 \sqrt{6} - 12 (\frac{\sqrt{216}}{3} - \frac{0}{1}))$

Step 9.2.3.12.2.4

Divide $0$ by $1$ .

$p (\frac{\sqrt{7776}}{5} + 36 \sqrt{6} - 12 (\frac{\sqrt{216}}{3} - 0))$

Step 9.2.3.13

Multiply $- 1$ by $0$ .

$p (\frac{\sqrt{7776}}{5} + 36 \sqrt{6} - 12 (\frac{\sqrt{216}}{3} + 0))$

Step 9.2.3.14

Add $\frac{\sqrt{216}}{3}$ and $0$ .

$p (\frac{\sqrt{7776}}{5} + 36 \sqrt{6} - 12 \frac{\sqrt{216}}{3})$

Step 9.2.3.15

Combine $- 12$ and $\frac{\sqrt{216}}{3}$ .

$p (\frac{\sqrt{7776}}{5} + 36 \sqrt{6} + \frac{- 12 \sqrt{216}}{3})$

Step 9.2.3.16

Cancel the common factor of $- 12$ and $3$ .

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Step 9.2.3.16.1

Factor $3$ out of $- 12 \sqrt{216}$ .

$p (\frac{\sqrt{7776}}{5} + 36 \sqrt{6} + \frac{3 (- 4 \sqrt{216})}{3})$

Step 9.2.3.16.2

Cancel the common factors.

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Step 9.2.3.16.2.1

Factor $3$ out of $3$ .

$p (\frac{\sqrt{7776}}{5} + 36 \sqrt{6} + \frac{3 (- 4 \sqrt{216})}{3 (1)})$

Step 9.2.3.16.2.2

Cancel the common factor.

$p (\frac{\sqrt{7776}}{5} + 36 \sqrt{6} + \frac{3 (- 4 \sqrt{216})}{3 \cdot 1})$

Step 9.2.3.16.2.3

Rewrite the expression.

$p (\frac{\sqrt{7776}}{5} + 36 \sqrt{6} + \frac{- 4 \sqrt{216}}{1})$

Step 9.2.3.16.2.4

Divide $- 4 \sqrt{216}$ by $1$ .

$p (\frac{\sqrt{7776}}{5} + 36 \sqrt{6} - 4 \sqrt{216})$

Step 9.3

Simplify.

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Step 9.3.1

Rewrite $7776$ as $36^{2} \cdot 6$ .

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Step 9.3.1.1

Factor $1296$ out of $7776$ .

$p (\frac{\sqrt{1296 (6)}}{5} + 36 \sqrt{6} - 4 \sqrt{216})$

Step 9.3.1.2

Rewrite $1296$ as $36^{2}$ .

$p (\frac{\sqrt{36^{2} \cdot 6}}{5} + 36 \sqrt{6} - 4 \sqrt{216})$

Step 9.3.2

Pull terms out from under the radical.

$p (\frac{36 \sqrt{6}}{5} + 36 \sqrt{6} - 4 \sqrt{216})$

Step 9.3.3

Rewrite $216$ as $6^{2} \cdot 6$ .

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Step 9.3.3.1

Factor $36$ out of $216$ .

$p (\frac{36 \sqrt{6}}{5} + 36 \sqrt{6} - 4 \sqrt{36 (6)})$

Step 9.3.3.2

Rewrite $36$ as $6^{2}$ .

$p (\frac{36 \sqrt{6}}{5} + 36 \sqrt{6} - 4 \sqrt{6^{2} \cdot 6})$

Step 9.3.4

Pull terms out from under the radical.

$p (\frac{36 \sqrt{6}}{5} + 36 \sqrt{6} - 4 (6 \sqrt{6}))$

Step 9.3.5

To write $36 \sqrt{6}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$p (\frac{36 \sqrt{6}}{5} + 36 \sqrt{6} \cdot \frac{5}{5} - 4 (6 \sqrt{6}))$

Step 9.3.6

Combine $36 \sqrt{6}$ and $\frac{5}{5}$ .

$p (\frac{36 \sqrt{6}}{5} + \frac{36 \sqrt{6} \cdot 5}{5} - 4 (6 \sqrt{6}))$

Step 9.3.7

Combine the numerators over the common denominator.

$p (\frac{36 \sqrt{6} + 36 \sqrt{6} \cdot 5}{5} - 4 (6 \sqrt{6}))$

Step 9.3.8

Multiply $5$ by $36$ .

$p (\frac{36 \sqrt{6} + 180 \sqrt{6}}{5} - 4 (6 \sqrt{6}))$

Step 9.3.9

Add $36 \sqrt{6}$ and $180 \sqrt{6}$ .

$p (\frac{216 \sqrt{6}}{5} - 4 (6 \sqrt{6}))$

Step 9.3.10

Multiply $6$ by $- 4$ .

$p (\frac{216 \sqrt{6}}{5} - 24 \sqrt{6})$

Step 9.3.11

To write $- 24 \sqrt{6}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$p (\frac{216 \sqrt{6}}{5} - 24 \sqrt{6} \cdot \frac{5}{5})$

Step 9.3.12

Combine $- 24 \sqrt{6}$ and $\frac{5}{5}$ .

$p (\frac{216 \sqrt{6}}{5} + \frac{- 24 \sqrt{6} \cdot 5}{5})$

Step 9.3.13

Combine the numerators over the common denominator.

$p \frac{216 \sqrt{6} - 24 \sqrt{6} \cdot 5}{5}$

Step 9.3.14

Multiply $5$ by $- 24$ .

$p \frac{216 \sqrt{6} - 120 \sqrt{6}}{5}$

Step 9.3.15

Subtract $120 \sqrt{6}$ from $216 \sqrt{6}$ .

$p \frac{96 \sqrt{6}}{5}$

Step 9.3.16

Combine $p$ and $\frac{96 \sqrt{6}}{5}$ .

$\frac{p (96 \sqrt{6})}{5}$

Step 9.3.17

Move $96$ to the left of $p$ .

$\frac{96 p \sqrt{6}}{5}$

Step 9.4

Reorder terms.

$\frac{96 \sqrt{6}}{5} p$

Step 10

Combine $\frac{96 \sqrt{6}}{5}$ and $p$ .

$\frac{96 \sqrt{6} p}{5}$

Step 11