Calculus Examples

$f (x) = \sqrt[3]{x}$ ; $[1, 8]$

Step 1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 2

$f (x)$ is continuous on $[1, 8]$ .

$f (x)$ is continuous

Step 3

The average value of function $f$ over the interval $[a, b]$ is defined as $A (x) = \frac{1}{b - a} \int_{a}^{b} f (x) d x$ .

$A (x) = \frac{1}{b - a} \int_{a}^{b} f (x) d x$

Step 4

Substitute the actual values into the formula for the average value of a function.

$A (x) = \frac{1}{8 - 1} (\int_{1}^{8} \sqrt[3]{x} d x)$

Step 5

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

$A (x) = \frac{1}{8 - 1} (\int_{1}^{8} x^{\frac{1}{3}} d x)$

Step 6

By the Power Rule, the integral of $x^{\frac{1}{3}}$ with respect to $x$ is $\frac{3}{4} x^{\frac{4}{3}}$ .

$A (x) = \frac{1}{8 - 1} (\frac{3}{4} x^{\frac{4}{3}}]_{1}^{8})$

Step 7

Substitute and simplify.

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Step 7.1

Evaluate $\frac{3}{4} x^{\frac{4}{3}}$ at $8$ and at $1$ .

$A (x) = \frac{1}{8 - 1} ((\frac{3}{4} \cdot 8^{\frac{4}{3}}) - \frac{3}{4} \cdot 1^{\frac{4}{3}})$

Step 7.2

Simplify.

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Step 7.2.1

Rewrite $8$ as $2^{3}$ .

$A (x) = \frac{1}{8 - 1} (\frac{3}{4} \cdot {(2^{3})}^{\frac{4}{3}} - \frac{3}{4} \cdot 1^{\frac{4}{3}})$

Step 7.2.2

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

$A (x) = \frac{1}{8 - 1} (\frac{3}{4} \cdot 2^{3 (\frac{4}{3})} - \frac{3}{4} \cdot 1^{\frac{4}{3}})$

Step 7.2.3

Cancel the common factor of $3$ .

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Step 7.2.3.1

Cancel the common factor.

$A (x) = \frac{1}{8 - 1} (\frac{3}{4} \cdot 2^{3 (\frac{4}{3})} - \frac{3}{4} \cdot 1^{\frac{4}{3}})$

Step 7.2.3.2

Rewrite the expression.

$A (x) = \frac{1}{8 - 1} (\frac{3}{4} \cdot 2^{4} - \frac{3}{4} \cdot 1^{\frac{4}{3}})$

Step 7.2.4

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

$A (x) = \frac{1}{8 - 1} (\frac{3}{4} \cdot 16 - \frac{3}{4} \cdot 1^{\frac{4}{3}})$

Step 7.2.5

Combine $\frac{3}{4}$ and $16$ .

$A (x) = \frac{1}{8 - 1} (\frac{3 \cdot 16}{4} - \frac{3}{4} \cdot 1^{\frac{4}{3}})$

Step 7.2.6

Multiply $3$ by $16$ .

$A (x) = \frac{1}{8 - 1} (\frac{48}{4} - \frac{3}{4} \cdot 1^{\frac{4}{3}})$

Step 7.2.7

Cancel the common factor of $48$ and $4$ .

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Step 7.2.7.1

Factor $4$ out of $48$ .

$A (x) = \frac{1}{8 - 1} (\frac{4 \cdot 12}{4} - \frac{3}{4} \cdot 1^{\frac{4}{3}})$

Step 7.2.7.2

Cancel the common factors.

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Step 7.2.7.2.1

Factor $4$ out of $4$ .

$A (x) = \frac{1}{8 - 1} (\frac{4 \cdot 12}{4 (1)} - \frac{3}{4} \cdot 1^{\frac{4}{3}})$

Step 7.2.7.2.2

Cancel the common factor.

$A (x) = \frac{1}{8 - 1} (\frac{4 \cdot 12}{4 \cdot 1} - \frac{3}{4} \cdot 1^{\frac{4}{3}})$

Step 7.2.7.2.3

Rewrite the expression.

$A (x) = \frac{1}{8 - 1} (\frac{12}{1} - \frac{3}{4} \cdot 1^{\frac{4}{3}})$

Step 7.2.7.2.4

Divide $12$ by $1$ .

$A (x) = \frac{1}{8 - 1} (12 - \frac{3}{4} \cdot 1^{\frac{4}{3}})$

Step 7.2.8

One to any power is one.

$A (x) = \frac{1}{8 - 1} (12 - \frac{3}{4} \cdot 1)$

Step 7.2.9

Multiply $- 1$ by $1$ .

$A (x) = \frac{1}{8 - 1} (12 - \frac{3}{4})$

Step 7.2.10

To write $12$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$A (x) = \frac{1}{8 - 1} (12 \cdot \frac{4}{4} - \frac{3}{4})$

Step 7.2.11

Combine $12$ and $\frac{4}{4}$ .

$A (x) = \frac{1}{8 - 1} (\frac{12 \cdot 4}{4} - \frac{3}{4})$

Step 7.2.12

Combine the numerators over the common denominator.

$A (x) = \frac{1}{8 - 1} (\frac{12 \cdot 4 - 3}{4})$

Step 7.2.13

Simplify the numerator.

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Step 7.2.13.1

Multiply $12$ by $4$ .

$A (x) = \frac{1}{8 - 1} (\frac{48 - 3}{4})$

Step 7.2.13.2

Subtract $3$ from $48$ .

$A (x) = \frac{1}{8 - 1} (\frac{45}{4})$

Step 8

Subtract $1$ from $8$ .

$A (x) = \frac{1}{7} \cdot \frac{45}{4}$

Step 9

Multiply $\frac{1}{7} \cdot \frac{45}{4}$ .

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Step 9.1

Multiply $\frac{1}{7}$ by $\frac{45}{4}$ .

$A (x) = \frac{45}{7 \cdot 4}$

Step 9.2

Multiply $7$ by $4$ .

$A (x) = \frac{45}{28}$

Step 10