Calculus Examples

$f (x) = x^{2} - 13$ , $[0, \sqrt{26}]$

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 $[0, \sqrt{26}]$ .

$f (x)$ 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.

$A (x) = \frac{1}{\sqrt{26} - 0} (\int_{0}^{\sqrt{26}} x^{2} - 13 d x)$

Step 5

Split the single integral into multiple integrals.

$A (x) = \frac{1}{\sqrt{26} - 0} (\int_{0}^{\sqrt{26}} x^{2} d x + \int_{0}^{\sqrt{26}} - 13 d x)$

Step 6

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

$A (x) = \frac{1}{\sqrt{26} - 0} (\frac{1}{3} x^{3}]_{0}^{\sqrt{26}} + \int_{0}^{\sqrt{26}} - 13 d x)$

Step 7

Apply the constant rule.

$A (x) = \frac{1}{\sqrt{26} - 0} (\frac{1}{3} x^{3}]_{0}^{\sqrt{26}} + - 13 x]_{0}^{\sqrt{26}})$

Step 8

Simplify the answer.

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Step 8.1

Combine $\frac{1}{3} x^{3}]_{0}^{\sqrt{26}}$ and $- 13 x]_{0}^{\sqrt{26}}$ .

$A (x) = \frac{1}{\sqrt{26} - 0} (\frac{1}{3} x^{3} - 13 x]_{0}^{\sqrt{26}})$

Step 8.2

Substitute and simplify.

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Step 8.2.1

Evaluate $\frac{1}{3} x^{3} - 13 x$ at $\sqrt{26}$ and at $0$ .

$A (x) = \frac{1}{\sqrt{26} - 0} ((\frac{1}{3} \cdot {\sqrt{26}}^{3} - 13 \sqrt{26}) - (\frac{1}{3} \cdot 0^{3} - 13 \cdot 0))$

Step 8.2.2

Simplify.

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Step 8.2.2.1

Rewrite ${\sqrt{26}}^{3}$ as $\sqrt{26^{3}}$ .

$A (x) = \frac{1}{\sqrt{26} - 0} (\frac{1}{3} \cdot \sqrt{26^{3}} - 13 \sqrt{26} - (\frac{1}{3} \cdot 0^{3} - 13 \cdot 0))$

Step 8.2.2.2

Raise $26$ to the power of $3$ .

$A (x) = \frac{1}{\sqrt{26} - 0} (\frac{1}{3} \cdot \sqrt{17576} - 13 \sqrt{26} - (\frac{1}{3} \cdot 0^{3} - 13 \cdot 0))$

Step 8.2.2.3

Combine $\frac{1}{3}$ and $\sqrt{17576}$ .

$A (x) = \frac{1}{\sqrt{26} - 0} (\frac{\sqrt{17576}}{3} - 13 \sqrt{26} - (\frac{1}{3} \cdot 0^{3} - 13 \cdot 0))$

Step 8.2.2.4

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

$A (x) = \frac{1}{\sqrt{26} - 0} (\frac{\sqrt{17576}}{3} - 13 \sqrt{26} - (\frac{1}{3} \cdot 0 - 13 \cdot 0))$

Step 8.2.2.5

Multiply $\frac{1}{3}$ by $0$ .

$A (x) = \frac{1}{\sqrt{26} - 0} (\frac{\sqrt{17576}}{3} - 13 \sqrt{26} - (0 - 13 \cdot 0))$

Step 8.2.2.6

Multiply $- 13$ by $0$ .

$A (x) = \frac{1}{\sqrt{26} - 0} (\frac{\sqrt{17576}}{3} - 13 \sqrt{26} - (0 + 0))$

Step 8.2.2.7

Add $0$ and $0$ .

$A (x) = \frac{1}{\sqrt{26} - 0} (\frac{\sqrt{17576}}{3} - 13 \sqrt{26} - 0)$

Step 8.2.2.8

Multiply $- 1$ by $0$ .

$A (x) = \frac{1}{\sqrt{26} - 0} (\frac{\sqrt{17576}}{3} - 13 \sqrt{26} + 0)$

Step 8.2.2.9

Add $\frac{\sqrt{17576}}{3} - 13 \sqrt{26}$ and $0$ .

$A (x) = \frac{1}{\sqrt{26} - 0} (\frac{\sqrt{17576}}{3} - 13 \sqrt{26})$

Step 8.3

Simplify.

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Step 8.3.1

Rewrite $17576$ as $26^{2} \cdot 26$ .

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Step 8.3.1.1

Factor $676$ out of $17576$ .

$A (x) = \frac{1}{\sqrt{26} - 0} (\frac{\sqrt{676 (26)}}{3} - 13 \sqrt{26})$

Step 8.3.1.2

Rewrite $676$ as $26^{2}$ .

$A (x) = \frac{1}{\sqrt{26} - 0} (\frac{\sqrt{26^{2} \cdot 26}}{3} - 13 \sqrt{26})$

Step 8.3.2

Pull terms out from under the radical.

$A (x) = \frac{1}{\sqrt{26} - 0} (\frac{26 \sqrt{26}}{3} - 13 \sqrt{26})$

Step 8.3.3

To write $- 13 \sqrt{26}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$A (x) = \frac{1}{\sqrt{26} - 0} (\frac{26 \sqrt{26}}{3} - 13 \sqrt{26} \cdot \frac{3}{3})$

Step 8.3.4

Combine $- 13 \sqrt{26}$ and $\frac{3}{3}$ .

$A (x) = \frac{1}{\sqrt{26} - 0} (\frac{26 \sqrt{26}}{3} + \frac{- 13 \sqrt{26} \cdot 3}{3})$

Step 8.3.5

Combine the numerators over the common denominator.

$A (x) = \frac{1}{\sqrt{26} - 0} (\frac{26 \sqrt{26} - 13 \sqrt{26} \cdot 3}{3})$

Step 8.3.6

Multiply $3$ by $- 13$ .

$A (x) = \frac{1}{\sqrt{26} - 0} (\frac{26 \sqrt{26} - 39 \sqrt{26}}{3})$

Step 8.3.7

Subtract $39 \sqrt{26}$ from $26 \sqrt{26}$ .

$A (x) = \frac{1}{\sqrt{26} - 0} (\frac{- 13 \sqrt{26}}{3})$

Step 8.3.8

Move the negative in front of the fraction.

$A (x) = \frac{1}{\sqrt{26} - 0} (- \frac{13 \sqrt{26}}{3})$

Step 9

Simplify the denominator.

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Step 9.1

Multiply $- 1$ by $0$ .

$A (x) = \frac{1}{\sqrt{26} + 0} \cdot (- \frac{13 \sqrt{26}}{3})$

Step 9.2

Add $\sqrt{26}$ and $0$ .

$A (x) = \frac{1}{\sqrt{26}} \cdot (- \frac{13 \sqrt{26}}{3})$

Step 10

Reduce the expression by cancelling the common factors.

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Step 10.1

Cancel the common factor of $\sqrt{26}$ .

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Step 10.1.1

Move the leading negative in $- \frac{13 \sqrt{26}}{3}$ into the numerator.

$A (x) = \frac{1}{\sqrt{26}} \cdot \frac{- 13 \sqrt{26}}{3}$

Step 10.1.2

Factor $\sqrt{26}$ out of $- 13 \sqrt{26}$ .

$A (x) = \frac{1}{\sqrt{26}} \cdot \frac{\sqrt{26} \cdot - 13}{3}$

Step 10.1.3

Cancel the common factor.

$A (x) = \frac{1}{\sqrt{26}} \cdot \frac{\sqrt{26} \cdot - 13}{3}$

Step 10.1.4

Rewrite the expression.

$A (x) = \frac{- 13}{3}$

Step 10.2

Move the negative in front of the fraction.

$A (x) = - \frac{13}{3}$

Step 11