Calculus Examples

$y = x^{4}$ , $[- 1, 1]$

Step 1

The Root Mean Square (RMS) of a function $f$ over a specified interval $[a, b]$ is the square root of the arithmetic mean (average) of the squares of the original values.

$f_{rms} = \sqrt{\frac{1}{b - a} \cdot \int_{a}^{b} f {(x)}^{2} d x}$

Step 2

Substitute the actual values into the formula for the root mean square of a function.

$f_{rms} = \sqrt{\frac{1}{1 + 1} \cdot (\int_{- 1}^{1} {(x^{4})}^{2} d x)}$

Step 3

Evaluate the integral.

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Step 3.1

Multiply the exponents in ${(x^{4})}^{2}$ .

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Step 3.1.1

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

$\int_{- 1}^{1} x^{4 \cdot 2} d x$

Step 3.1.2

Multiply $4$ by $2$ .

$\int_{- 1}^{1} x^{8} d x$

Step 3.2

By the Power Rule, the integral of $x^{8}$ with respect to $x$ is $\frac{1}{9} x^{9}$ .

$\frac{1}{9} x^{9}]_{- 1}^{1}$

Step 3.3

Substitute and simplify.

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Step 3.3.1

Evaluate $\frac{1}{9} x^{9}$ at $1$ and at $- 1$ .

$(\frac{1}{9} \cdot 1^{9}) - \frac{1}{9} {(- 1)}^{9}$

Step 3.3.2

Simplify.

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Step 3.3.2.1

One to any power is one.

$\frac{1}{9} \cdot 1 - \frac{1}{9} {(- 1)}^{9}$

Step 3.3.2.2

Multiply $\frac{1}{9}$ by $1$ .

$\frac{1}{9} - \frac{1}{9} {(- 1)}^{9}$

Step 3.3.2.3

Raise $- 1$ to the power of $9$ .

$\frac{1}{9} - \frac{1}{9} \cdot - 1$

Step 3.3.2.4

Multiply $- 1$ by $- 1$ .

$\frac{1}{9} + 1 (\frac{1}{9})$

Step 3.3.2.5

Multiply $\frac{1}{9}$ by $1$ .

$\frac{1}{9} + \frac{1}{9}$

Step 3.3.2.6

Combine the numerators over the common denominator.

$\frac{1 + 1}{9}$

Step 3.3.2.7

Add $1$ and $1$ .

$\frac{2}{9}$

Step 4

Simplify the root mean square formula.

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Step 4.1

Multiply $\frac{1}{1 + 1}$ by $\frac{2}{9}$ .

$f_{rms} = \sqrt{\frac{2}{(1 + 1) \cdot 9}}$

Step 4.2

Add $1$ and $1$ .

$f_{rms} = \sqrt{\frac{2}{2 \cdot 9}}$

Step 4.3

Reduce the expression $\frac{2}{2 \cdot 9}$ by cancelling the common factors.

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Step 4.3.1

Cancel the common factor.

$f_{rms} = \sqrt{\frac{2}{2 \cdot 9}}$

Step 4.3.2

Rewrite the expression.

$f_{rms} = \sqrt{\frac{1}{9}}$

Step 4.4

Rewrite $\sqrt{\frac{1}{9}}$ as $\frac{\sqrt{1}}{\sqrt{9}}$ .

$f_{rms} = \frac{\sqrt{1}}{\sqrt{9}}$

Step 4.5

Any root of $1$ is $1$ .

$f_{rms} = \frac{1}{\sqrt{9}}$

Step 4.6

Simplify the denominator.

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Step 4.6.1

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

$f_{rms} = \frac{1}{\sqrt{3^{2}}}$

Step 4.6.2

Pull terms out from under the radical, assuming positive real numbers.

$f_{rms} = \frac{1}{3}$

Step 5