Calculus Examples

$2 p \int_{0}^{1.67332005} (y) (2.8 y - y^{3}) d y$

Step 1

Expand $y (2.8 y - y^{3})$ .

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Step 1.1

Apply the distributive property.

$2 p \int_{0}^{1.67332005} y (2.8 y) + y (- y^{3}) d y$

Step 1.2

Reorder $y$ and $2.8$ .

$2 p \int_{0}^{1.67332005} 2.8 \cdot y \cdot y + y (- y^{3}) d y$

Step 1.3

Reorder $y$ and $- 1$ .

$2 p \int_{0}^{1.67332005} 2.8 \cdot y \cdot y - 1 \cdot y \cdot y^{3} d y$

Step 1.4

Raise $y$ to the power of $1$ .

$2 p \int_{0}^{1.67332005} 2.8 (y^{1} y) - 1 y \cdot y^{3} d y$

Step 1.5

Raise $y$ to the power of $1$ .

$2 p \int_{0}^{1.67332005} 2.8 (y^{1} y^{1}) - 1 y \cdot y^{3} d y$

Step 1.6

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

$2 p \int_{0}^{1.67332005} 2.8 y^{1 + 1} - 1 y \cdot y^{3} d y$

Step 1.7

Add $1$ and $1$ .

$2 p \int_{0}^{1.67332005} 2.8 y^{2} - 1 y \cdot y^{3} d y$

Step 1.8

Factor out negative.

$2 p \int_{0}^{1.67332005} 2.8 y^{2} - (y \cdot y^{3}) d y$

Step 1.9

Raise $y$ to the power of $1$ .

$2 p \int_{0}^{1.67332005} 2.8 y^{2} - (y^{1} y^{3}) d y$

Step 1.10

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

$2 p \int_{0}^{1.67332005} 2.8 y^{2} - y^{1 + 3} d y$

Step 1.11

Add $1$ and $3$ .

$2 p \int_{0}^{1.67332005} 2.8 y^{2} - y^{4} d y$

Step 1.12

Reorder $2.8 y^{2}$ and $- y^{4}$ .

$2 p \int_{0}^{1.67332005} - y^{4} + 2.8 y^{2} d y$

Step 2

Split the single integral into multiple integrals.

$2 p (\int_{0}^{1.67332005} - y^{4} d y + \int_{0}^{1.67332005} 2.8 y^{2} d y)$

Step 3

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

$2 p (- \int_{0}^{1.67332005} y^{4} d y + \int_{0}^{1.67332005} 2.8 y^{2} d y)$

Step 4

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

$2 p (- (\frac{1}{5} y^{5}]_{0}^{1.67332005}) + \int_{0}^{1.67332005} 2.8 y^{2} d y)$

Step 5

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

$2 p (- (\frac{y^{5}}{5}]_{0}^{1.67332005}) + \int_{0}^{1.67332005} 2.8 y^{2} d y)$

Step 6

Since $2.8$ is constant with respect to $y$ , move $2.8$ out of the integral.

$2 p (- (\frac{y^{5}}{5}]_{0}^{1.67332005}) + 2.8 \int_{0}^{1.67332005} y^{2} d y)$

Step 7

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

$2 p (- (\frac{y^{5}}{5}]_{0}^{1.67332005}) + 2.8 (\frac{1}{3} y^{3}]_{0}^{1.67332005}))$

Step 8

Simplify the answer.

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Step 8.1

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

$2 p (- (\frac{y^{5}}{5}]_{0}^{1.67332005}) + 2.8 (\frac{y^{3}}{3}]_{0}^{1.67332005}))$

Step 8.2

Substitute and simplify.

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Step 8.2.1

Evaluate $\frac{y^{5}}{5}$ at $1.67332005$ and at $0$ .

$2 p (- ((\frac{{1.67332005}^{5}}{5}) - \frac{0^{5}}{5}) + 2.8 (\frac{y^{3}}{3}]_{0}^{1.67332005}))$

Step 8.2.2

Evaluate $\frac{y^{3}}{3}$ at $1.67332005$ and at $0$ .

$2 p (- (\frac{{1.67332005}^{5}}{5} - \frac{0^{5}}{5}) + 2.8 (\frac{{1.67332005}^{3}}{3} - \frac{0^{3}}{3}))$

Step 8.2.3

Simplify.

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Step 8.2.3.1

Raise $1.67332005$ to the power of $5$ .

$2 p (- (\frac{13.11882909}{5} - \frac{0^{5}}{5}) + 2.8 (\frac{{1.67332005}^{3}}{3} - \frac{0^{3}}{3}))$

Step 8.2.3.2

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

$2 p (- (\frac{13.11882909}{5} - \frac{0}{5}) + 2.8 (\frac{{1.67332005}^{3}}{3} - \frac{0^{3}}{3}))$

Step 8.2.3.3

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

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Step 8.2.3.3.1

Factor $5$ out of $0$ .

$2 p (- (\frac{13.11882909}{5} - \frac{5 (0)}{5}) + 2.8 (\frac{{1.67332005}^{3}}{3} - \frac{0^{3}}{3}))$

Step 8.2.3.3.2

Cancel the common factors.

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Step 8.2.3.3.2.1

Factor $5$ out of $5$ .

$2 p (- (\frac{13.11882909}{5} - \frac{5 \cdot 0}{5 \cdot 1}) + 2.8 (\frac{{1.67332005}^{3}}{3} - \frac{0^{3}}{3}))$

Step 8.2.3.3.2.2

Cancel the common factor.

$2 p (- (\frac{13.11882909}{5} - \frac{5 \cdot 0}{5 \cdot 1}) + 2.8 (\frac{{1.67332005}^{3}}{3} - \frac{0^{3}}{3}))$

Step 8.2.3.3.2.3

Rewrite the expression.

$2 p (- (\frac{13.11882909}{5} - \frac{0}{1}) + 2.8 (\frac{{1.67332005}^{3}}{3} - \frac{0^{3}}{3}))$

Step 8.2.3.3.2.4

Divide $0$ by $1$ .

$2 p (- (\frac{13.11882909}{5} - 0) + 2.8 (\frac{{1.67332005}^{3}}{3} - \frac{0^{3}}{3}))$

Step 8.2.3.4

Multiply $- 1$ by $0$ .

$2 p (- (\frac{13.11882909}{5} + 0) + 2.8 (\frac{{1.67332005}^{3}}{3} - \frac{0^{3}}{3}))$

Step 8.2.3.5

Add $\frac{13.11882909}{5}$ and $0$ .

$2 p (- \frac{13.11882909}{5} + 2.8 (\frac{{1.67332005}^{3}}{3} - \frac{0^{3}}{3}))$

Step 8.2.3.6

Raise $1.67332005$ to the power of $3$ .

$2 p (- \frac{13.11882909}{5} + 2.8 (\frac{4.68529612}{3} - \frac{0^{3}}{3}))$

Step 8.2.3.7

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

$2 p (- \frac{13.11882909}{5} + 2.8 (\frac{4.68529612}{3} - \frac{0}{3}))$

Step 8.2.3.8

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

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Step 8.2.3.8.1

Factor $3$ out of $0$ .

$2 p (- \frac{13.11882909}{5} + 2.8 (\frac{4.68529612}{3} - \frac{3 (0)}{3}))$

Step 8.2.3.8.2

Cancel the common factors.

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Step 8.2.3.8.2.1

Factor $3$ out of $3$ .

$2 p (- \frac{13.11882909}{5} + 2.8 (\frac{4.68529612}{3} - \frac{3 \cdot 0}{3 \cdot 1}))$

Step 8.2.3.8.2.2

Cancel the common factor.

$2 p (- \frac{13.11882909}{5} + 2.8 (\frac{4.68529612}{3} - \frac{3 \cdot 0}{3 \cdot 1}))$

Step 8.2.3.8.2.3

Rewrite the expression.

$2 p (- \frac{13.11882909}{5} + 2.8 (\frac{4.68529612}{3} - \frac{0}{1}))$

Step 8.2.3.8.2.4

Divide $0$ by $1$ .

$2 p (- \frac{13.11882909}{5} + 2.8 (\frac{4.68529612}{3} - 0))$

Step 8.2.3.9

Multiply $- 1$ by $0$ .

$2 p (- \frac{13.11882909}{5} + 2.8 (\frac{4.68529612}{3} + 0))$

Step 8.2.3.10

Add $\frac{4.68529612}{3}$ and $0$ .

$2 p (- \frac{13.11882909}{5} + 2.8 (\frac{4.68529612}{3}))$

Step 8.2.3.11

Combine $2.8$ and $\frac{4.68529612}{3}$ .

$2 p (- \frac{13.11882909}{5} + \frac{2.8 \cdot 4.68529612}{3})$

Step 8.2.3.12

Multiply $2.8$ by $4.68529612$ .

$2 p (- \frac{13.11882909}{5} + \frac{13.11882914}{3})$

Step 8.2.3.13

To write $- \frac{13.11882909}{5}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$2 p (- \frac{13.11882909}{5} \cdot \frac{3}{3} + \frac{13.11882914}{3})$

Step 8.2.3.14

To write $\frac{13.11882914}{3}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$2 p (- \frac{13.11882909}{5} \cdot \frac{3}{3} + \frac{13.11882914}{3} \cdot \frac{5}{5})$

Step 8.2.3.15

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.2.3.15.1

Multiply $\frac{13.11882909}{5}$ by $\frac{3}{3}$ .

$2 p (- \frac{13.11882909 \cdot 3}{5 \cdot 3} + \frac{13.11882914}{3} \cdot \frac{5}{5})$

Step 8.2.3.15.2

Multiply $5$ by $3$ .

$2 p (- \frac{13.11882909 \cdot 3}{15} + \frac{13.11882914}{3} \cdot \frac{5}{5})$

Step 8.2.3.15.3

Multiply $\frac{13.11882914}{3}$ by $\frac{5}{5}$ .

$2 p (- \frac{13.11882909 \cdot 3}{15} + \frac{13.11882914 \cdot 5}{3 \cdot 5})$

Step 8.2.3.15.4

Multiply $3$ by $5$ .

$2 p (- \frac{13.11882909 \cdot 3}{15} + \frac{13.11882914 \cdot 5}{15})$

Step 8.2.3.16

Combine the numerators over the common denominator.

$2 p \frac{- 13.11882909 \cdot 3 + 13.11882914 \cdot 5}{15}$

Step 8.2.3.17

Multiply $- 13.11882909$ by $3$ .

$2 p \frac{- 39.35648728 + 13.11882914 \cdot 5}{15}$

Step 8.2.3.18

Multiply $13.11882914$ by $5$ .

$2 p \frac{- 39.35648728 + 65.59414571}{15}$

Step 8.2.3.19

Add $- 39.35648728$ and $65.59414571$ .

$2 p \frac{26.23765843}{15}$

Step 8.2.3.20

Combine $2$ and $\frac{26.23765843}{15}$ .

$p \frac{2 \cdot 26.23765843}{15}$

Step 8.2.3.21

Multiply $2$ by $26.23765843$ .

$p \frac{52.47531686}{15}$

Step 8.2.3.22

Combine $p$ and $\frac{52.47531686}{15}$ .

$\frac{p \cdot 52.47531686}{15}$

Step 8.2.3.23

Move $52.47531686$ to the left of $p$ .

$\frac{52.47531686 p}{15}$

Step 8.3

Reorder terms.

$\frac{52.47531686}{15} p$

Step 9