Calculus Examples

$\int_{4}^{9} \sqrt{x} d x$

Step 1

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

$\int_{4}^{9} x^{\frac{1}{2}} d x$

Step 2

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

$\frac{2}{3} x^{\frac{3}{2}}]_{4}^{9}$

Step 3

Substitute and simplify.

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Step 3.1

Evaluate $\frac{2}{3} x^{\frac{3}{2}}$ at $9$ and at $4$ .

$(\frac{2}{3} \cdot 9^{\frac{3}{2}}) - \frac{2}{3} \cdot 4^{\frac{3}{2}}$

Step 3.2

Simplify.

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Step 3.2.1

Rewrite $9$ as $3^{2}$ .

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

Step 3.2.2

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

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

Step 3.2.3

Cancel the common factor of $2$ .

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Step 3.2.3.1

Cancel the common factor.

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

Step 3.2.3.2

Rewrite the expression.

$\frac{2}{3} \cdot 3^{3} - \frac{2}{3} \cdot 4^{\frac{3}{2}}$

Step 3.2.4

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

$\frac{2}{3} \cdot 27 - \frac{2}{3} \cdot 4^{\frac{3}{2}}$

Step 3.2.5

Combine $\frac{2}{3}$ and $27$ .

$\frac{2 \cdot 27}{3} - \frac{2}{3} \cdot 4^{\frac{3}{2}}$

Step 3.2.6

Multiply $2$ by $27$ .

$\frac{54}{3} - \frac{2}{3} \cdot 4^{\frac{3}{2}}$

Step 3.2.7

Cancel the common factor of $54$ and $3$ .

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Step 3.2.7.1

Factor $3$ out of $54$ .

$\frac{3 \cdot 18}{3} - \frac{2}{3} \cdot 4^{\frac{3}{2}}$

Step 3.2.7.2

Cancel the common factors.

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Step 3.2.7.2.1

Factor $3$ out of $3$ .

$\frac{3 \cdot 18}{3 (1)} - \frac{2}{3} \cdot 4^{\frac{3}{2}}$

Step 3.2.7.2.2

Cancel the common factor.

$\frac{3 \cdot 18}{3 \cdot 1} - \frac{2}{3} \cdot 4^{\frac{3}{2}}$

Step 3.2.7.2.3

Rewrite the expression.

$\frac{18}{1} - \frac{2}{3} \cdot 4^{\frac{3}{2}}$

Step 3.2.7.2.4

Divide $18$ by $1$ .

$18 - \frac{2}{3} \cdot 4^{\frac{3}{2}}$

Step 3.2.8

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

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

Step 3.2.9

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

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

Step 3.2.10

Cancel the common factor of $2$ .

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Step 3.2.10.1

Cancel the common factor.

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

Step 3.2.10.2

Rewrite the expression.

$18 - \frac{2}{3} \cdot 2^{3}$

Step 3.2.11

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

$18 - \frac{2}{3} \cdot 8$

Step 3.2.12

Multiply $8$ by $- 1$ .

$18 - 8 (\frac{2}{3})$

Step 3.2.13

Combine $- 8$ and $\frac{2}{3}$ .

$18 + \frac{- 8 \cdot 2}{3}$

Step 3.2.14

Multiply $- 8$ by $2$ .

$18 + \frac{- 16}{3}$

Step 3.2.15

Move the negative in front of the fraction.

$18 - \frac{16}{3}$

Step 3.2.16

To write $18$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$18 \cdot \frac{3}{3} - \frac{16}{3}$

Step 3.2.17

Combine $18$ and $\frac{3}{3}$ .

$\frac{18 \cdot 3}{3} - \frac{16}{3}$

Step 3.2.18

Combine the numerators over the common denominator.

$\frac{18 \cdot 3 - 16}{3}$

Step 3.2.19

Simplify the numerator.

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Step 3.2.19.1

Multiply $18$ by $3$ .

$\frac{54 - 16}{3}$

Step 3.2.19.2

Subtract $16$ from $54$ .

$\frac{38}{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{38}{3}$

Decimal Form:

$12.\overline{6}$

Mixed Number Form:

$12 \frac{2}{3}$

Step 5