Calculus Examples

$\int_{0}^{4} (2 \sqrt{y} - y) d y$

Step 1

Remove parentheses.

$\int_{0}^{4} 2 \sqrt{y} - y d y$

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y}$ as $y^{\frac{1}{2}}$ .

$2 \int_{0}^{4} y^{\frac{1}{2}} d y + \int_{0}^{4} - y d y$

Step 5

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

$2 (\frac{2}{3} y^{\frac{3}{2}}]_{0}^{4}) + \int_{0}^{4} - y d y$

Step 6

Combine $\frac{2}{3}$ and $y^{\frac{3}{2}}$ .

$2 (\frac{2 y^{\frac{3}{2}}}{3}]_{0}^{4}) + \int_{0}^{4} - y d y$

Step 7

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

$2 (\frac{2 y^{\frac{3}{2}}}{3}]_{0}^{4}) - \int_{0}^{4} y d y$

Step 8

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

$2 (\frac{2 y^{\frac{3}{2}}}{3}]_{0}^{4}) - (\frac{1}{2} y^{2}]_{0}^{4})$

Step 9

Simplify the answer.

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Step 9.1

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

$2 (\frac{2 y^{\frac{3}{2}}}{3}]_{0}^{4}) - (\frac{y^{2}}{2}]_{0}^{4})$

Step 9.2

Substitute and simplify.

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Step 9.2.1

Evaluate $\frac{2 y^{\frac{3}{2}}}{3}$ at $4$ and at $0$ .

$2 ((\frac{2 \cdot 4^{\frac{3}{2}}}{3}) - \frac{2 \cdot 0^{\frac{3}{2}}}{3}) - (\frac{y^{2}}{2}]_{0}^{4})$

Step 9.2.2

Evaluate $\frac{y^{2}}{2}$ at $4$ and at $0$ .

$2 (\frac{2 \cdot 4^{\frac{3}{2}}}{3} - \frac{2 \cdot 0^{\frac{3}{2}}}{3}) - (\frac{4^{2}}{2} - \frac{0^{2}}{2})$

Step 9.2.3

Simplify.

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Step 9.2.3.1

Rewrite $4$ as $2^{2}$ .

$2 (\frac{2 \cdot {(2^{2})}^{\frac{3}{2}}}{3} - \frac{2 \cdot 0^{\frac{3}{2}}}{3}) - (\frac{4^{2}}{2} - \frac{0^{2}}{2})$

Step 9.2.3.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$2 (\frac{2 \cdot 2^{2 (\frac{3}{2})}}{3} - \frac{2 \cdot 0^{\frac{3}{2}}}{3}) - (\frac{4^{2}}{2} - \frac{0^{2}}{2})$

Step 9.2.3.3

Cancel the common factor of $2$ .

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Step 9.2.3.3.1

Cancel the common factor.

$2 (\frac{2 \cdot 2^{2 (\frac{3}{2})}}{3} - \frac{2 \cdot 0^{\frac{3}{2}}}{3}) - (\frac{4^{2}}{2} - \frac{0^{2}}{2})$

Step 9.2.3.3.2

Rewrite the expression.

$2 (\frac{2 \cdot 2^{3}}{3} - \frac{2 \cdot 0^{\frac{3}{2}}}{3}) - (\frac{4^{2}}{2} - \frac{0^{2}}{2})$

Step 9.2.3.4

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

$2 (\frac{2 \cdot 8}{3} - \frac{2 \cdot 0^{\frac{3}{2}}}{3}) - (\frac{4^{2}}{2} - \frac{0^{2}}{2})$

Step 9.2.3.5

Multiply $2$ by $8$ .

$2 (\frac{16}{3} - \frac{2 \cdot 0^{\frac{3}{2}}}{3}) - (\frac{4^{2}}{2} - \frac{0^{2}}{2})$

Step 9.2.3.6

Rewrite $0$ as $0^{2}$ .

$2 (\frac{16}{3} - \frac{2 \cdot {(0^{2})}^{\frac{3}{2}}}{3}) - (\frac{4^{2}}{2} - \frac{0^{2}}{2})$

Step 9.2.3.7

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$2 (\frac{16}{3} - \frac{2 \cdot 0^{2 (\frac{3}{2})}}{3}) - (\frac{4^{2}}{2} - \frac{0^{2}}{2})$

Step 9.2.3.8

Cancel the common factor of $2$ .

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Step 9.2.3.8.1

Cancel the common factor.

$2 (\frac{16}{3} - \frac{2 \cdot 0^{2 (\frac{3}{2})}}{3}) - (\frac{4^{2}}{2} - \frac{0^{2}}{2})$

Step 9.2.3.8.2

Rewrite the expression.

$2 (\frac{16}{3} - \frac{2 \cdot 0^{3}}{3}) - (\frac{4^{2}}{2} - \frac{0^{2}}{2})$

Step 9.2.3.9

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

$2 (\frac{16}{3} - \frac{2 \cdot 0}{3}) - (\frac{4^{2}}{2} - \frac{0^{2}}{2})$

Step 9.2.3.10

Multiply $2$ by $0$ .

$2 (\frac{16}{3} - \frac{0}{3}) - (\frac{4^{2}}{2} - \frac{0^{2}}{2})$

Step 9.2.3.11

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

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Step 9.2.3.11.1

Factor $3$ out of $0$ .

$2 (\frac{16}{3} - \frac{3 (0)}{3}) - (\frac{4^{2}}{2} - \frac{0^{2}}{2})$

Step 9.2.3.11.2

Cancel the common factors.

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Step 9.2.3.11.2.1

Factor $3$ out of $3$ .

$2 (\frac{16}{3} - \frac{3 \cdot 0}{3 \cdot 1}) - (\frac{4^{2}}{2} - \frac{0^{2}}{2})$

Step 9.2.3.11.2.2

Cancel the common factor.

$2 (\frac{16}{3} - \frac{3 \cdot 0}{3 \cdot 1}) - (\frac{4^{2}}{2} - \frac{0^{2}}{2})$

Step 9.2.3.11.2.3

Rewrite the expression.

$2 (\frac{16}{3} - \frac{0}{1}) - (\frac{4^{2}}{2} - \frac{0^{2}}{2})$

Step 9.2.3.11.2.4

Divide $0$ by $1$ .

$2 (\frac{16}{3} - 0) - (\frac{4^{2}}{2} - \frac{0^{2}}{2})$

Step 9.2.3.12

Multiply $- 1$ by $0$ .

$2 (\frac{16}{3} + 0) - (\frac{4^{2}}{2} - \frac{0^{2}}{2})$

Step 9.2.3.13

Add $\frac{16}{3}$ and $0$ .

$2 (\frac{16}{3}) - (\frac{4^{2}}{2} - \frac{0^{2}}{2})$

Step 9.2.3.14

Combine $2$ and $\frac{16}{3}$ .

$\frac{2 \cdot 16}{3} - (\frac{4^{2}}{2} - \frac{0^{2}}{2})$

Step 9.2.3.15

Multiply $2$ by $16$ .

$\frac{32}{3} - (\frac{4^{2}}{2} - \frac{0^{2}}{2})$

Step 9.2.3.16

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

$\frac{32}{3} - (\frac{16}{2} - \frac{0^{2}}{2})$

Step 9.2.3.17

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

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Step 9.2.3.17.1

Factor $2$ out of $16$ .

$\frac{32}{3} - (\frac{2 \cdot 8}{2} - \frac{0^{2}}{2})$

Step 9.2.3.17.2

Cancel the common factors.

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Step 9.2.3.17.2.1

Factor $2$ out of $2$ .

$\frac{32}{3} - (\frac{2 \cdot 8}{2 (1)} - \frac{0^{2}}{2})$

Step 9.2.3.17.2.2

Cancel the common factor.

$\frac{32}{3} - (\frac{2 \cdot 8}{2 \cdot 1} - \frac{0^{2}}{2})$

Step 9.2.3.17.2.3

Rewrite the expression.

$\frac{32}{3} - (\frac{8}{1} - \frac{0^{2}}{2})$

Step 9.2.3.17.2.4

Divide $8$ by $1$ .

$\frac{32}{3} - (8 - \frac{0^{2}}{2})$

Step 9.2.3.18

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

$\frac{32}{3} - (8 - \frac{0}{2})$

Step 9.2.3.19

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

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Step 9.2.3.19.1

Factor $2$ out of $0$ .

$\frac{32}{3} - (8 - \frac{2 (0)}{2})$

Step 9.2.3.19.2

Cancel the common factors.

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Step 9.2.3.19.2.1

Factor $2$ out of $2$ .

$\frac{32}{3} - (8 - \frac{2 \cdot 0}{2 \cdot 1})$

Step 9.2.3.19.2.2

Cancel the common factor.

$\frac{32}{3} - (8 - \frac{2 \cdot 0}{2 \cdot 1})$

Step 9.2.3.19.2.3

Rewrite the expression.

$\frac{32}{3} - (8 - \frac{0}{1})$

Step 9.2.3.19.2.4

Divide $0$ by $1$ .

$\frac{32}{3} - (8 - 0)$

Step 9.2.3.20

Multiply $- 1$ by $0$ .

$\frac{32}{3} - (8 + 0)$

Step 9.2.3.21

Add $8$ and $0$ .

$\frac{32}{3} - 1 \cdot 8$

Step 9.2.3.22

Multiply $- 1$ by $8$ .

$\frac{32}{3} - 8$

Step 9.2.3.23

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

$\frac{32}{3} - 8 \cdot \frac{3}{3}$

Step 9.2.3.24

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

$\frac{32}{3} + \frac{- 8 \cdot 3}{3}$

Step 9.2.3.25

Combine the numerators over the common denominator.

$\frac{32 - 8 \cdot 3}{3}$

Step 9.2.3.26

Simplify the numerator.

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Step 9.2.3.26.1

Multiply $- 8$ by $3$ .

$\frac{32 - 24}{3}$

Step 9.2.3.26.2

Subtract $24$ from $32$ .

$\frac{8}{3}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{8}{3}$

Decimal Form:

$2.\overline{6}$

Mixed Number Form:

$2 \frac{2}{3}$

Step 11