Calculus Examples

$\int_{0}^{16} 4 \sqrt{x} d x$

Step 1

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$4 \int_{0}^{16} \sqrt{x} d x$

Step 2

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

$4 \int_{0}^{16} x^{\frac{1}{2}} d x$

Step 3

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

$4 (\frac{2}{3} x^{\frac{3}{2}}]_{0}^{16})$

Step 4

Simplify the answer.

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Step 4.1

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

$4 (\frac{2 x^{\frac{3}{2}}}{3}]_{0}^{16})$

Step 4.2

Substitute and simplify.

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Step 4.2.1

Evaluate $\frac{2 x^{\frac{3}{2}}}{3}$ at $16$ and at $0$ .

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

Step 4.2.2

Simplify.

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Step 4.2.2.1

Rewrite $16$ as $4^{2}$ .

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

Step 4.2.2.2

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

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

Step 4.2.2.3

Cancel the common factor of $2$ .

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Step 4.2.2.3.1

Cancel the common factor.

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

Step 4.2.2.3.2

Rewrite the expression.

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

Step 4.2.2.4

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

$4 (\frac{2 \cdot 64}{3} - \frac{2 \cdot 0^{\frac{3}{2}}}{3})$

Step 4.2.2.5

Multiply $2$ by $64$ .

$4 (\frac{128}{3} - \frac{2 \cdot 0^{\frac{3}{2}}}{3})$

Step 4.2.2.6

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

$4 (\frac{128}{3} - \frac{2 \cdot {(0^{2})}^{\frac{3}{2}}}{3})$

Step 4.2.2.7

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

$4 (\frac{128}{3} - \frac{2 \cdot 0^{2 (\frac{3}{2})}}{3})$

Step 4.2.2.8

Cancel the common factor of $2$ .

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Step 4.2.2.8.1

Cancel the common factor.

$4 (\frac{128}{3} - \frac{2 \cdot 0^{2 (\frac{3}{2})}}{3})$

Step 4.2.2.8.2

Rewrite the expression.

$4 (\frac{128}{3} - \frac{2 \cdot 0^{3}}{3})$

Step 4.2.2.9

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

$4 (\frac{128}{3} - \frac{2 \cdot 0}{3})$

Step 4.2.2.10

Multiply $2$ by $0$ .

$4 (\frac{128}{3} - \frac{0}{3})$

Step 4.2.2.11

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

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Step 4.2.2.11.1

Factor $3$ out of $0$ .

$4 (\frac{128}{3} - \frac{3 (0)}{3})$

Step 4.2.2.11.2

Cancel the common factors.

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Step 4.2.2.11.2.1

Factor $3$ out of $3$ .

$4 (\frac{128}{3} - \frac{3 \cdot 0}{3 \cdot 1})$

Step 4.2.2.11.2.2

Cancel the common factor.

$4 (\frac{128}{3} - \frac{3 \cdot 0}{3 \cdot 1})$

Step 4.2.2.11.2.3

Rewrite the expression.

$4 (\frac{128}{3} - \frac{0}{1})$

Step 4.2.2.11.2.4

Divide $0$ by $1$ .

$4 (\frac{128}{3} - 0)$

Step 4.2.2.12

Multiply $- 1$ by $0$ .

$4 (\frac{128}{3} + 0)$

Step 4.2.2.13

Add $\frac{128}{3}$ and $0$ .

$4 (\frac{128}{3})$

Step 4.2.2.14

Combine $4$ and $\frac{128}{3}$ .

$\frac{4 \cdot 128}{3}$

Step 4.2.2.15

Multiply $4$ by $128$ .

$\frac{512}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{512}{3}$

Decimal Form:

$170.\overline{6}$

Mixed Number Form:

$170 \frac{2}{3}$

Step 6