Calculus Examples

$y = x^{2}$ , $[2, 5]$

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.

$f_{rms} = \sqrt{\frac{1}{5 - 2} \cdot (\int_{2}^{5} {(x^{2})}^{2} d x)}$

Step 3

Evaluate the integral.

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Step 3.1

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

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Step 3.1.1

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

$\int_{2}^{5} x^{2 \cdot 2} d x$

Step 3.1.2

Multiply $2$ by $2$ .

$\int_{2}^{5} x^{4} d x$

Step 3.2

By the Power Rule, the integral of $x^{4}$ with respect to $x$ is $\frac{1}{5} x^{5}$ .

$\frac{1}{5} x^{5}]_{2}^{5}$

Step 3.3

Substitute and simplify.

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Step 3.3.1

Evaluate $\frac{1}{5} x^{5}$ at $5$ and at $2$ .

$(\frac{1}{5} \cdot 5^{5}) - \frac{1}{5} \cdot 2^{5}$

Step 3.3.2

Simplify.

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Step 3.3.2.1

Raise $5$ to the power of $5$ .

$\frac{1}{5} \cdot 3125 - \frac{1}{5} \cdot 2^{5}$

Step 3.3.2.2

Combine $\frac{1}{5}$ and $3125$ .

$\frac{3125}{5} - \frac{1}{5} \cdot 2^{5}$

Step 3.3.2.3

Cancel the common factor of $3125$ and $5$ .

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Step 3.3.2.3.1

Factor $5$ out of $3125$ .

$\frac{5 \cdot 625}{5} - \frac{1}{5} \cdot 2^{5}$

Step 3.3.2.3.2

Cancel the common factors.

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Step 3.3.2.3.2.1

Factor $5$ out of $5$ .

$\frac{5 \cdot 625}{5 (1)} - \frac{1}{5} \cdot 2^{5}$

Step 3.3.2.3.2.2

Cancel the common factor.

$\frac{5 \cdot 625}{5 \cdot 1} - \frac{1}{5} \cdot 2^{5}$

Step 3.3.2.3.2.3

Rewrite the expression.

$\frac{625}{1} - \frac{1}{5} \cdot 2^{5}$

Step 3.3.2.3.2.4

Divide $625$ by $1$ .

$625 - \frac{1}{5} \cdot 2^{5}$

Step 3.3.2.4

Raise $2$ to the power of $5$ .

$625 - \frac{1}{5} \cdot 32$

Step 3.3.2.5

Multiply $32$ by $- 1$ .

$625 - 32 (\frac{1}{5})$

Step 3.3.2.6

Combine $- 32$ and $\frac{1}{5}$ .

$625 + \frac{- 32}{5}$

Step 3.3.2.7

Move the negative in front of the fraction.

$625 - \frac{32}{5}$

Step 3.3.2.8

To write $625$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$625 \cdot \frac{5}{5} - \frac{32}{5}$

Step 3.3.2.9

Combine $625$ and $\frac{5}{5}$ .

$\frac{625 \cdot 5}{5} - \frac{32}{5}$

Step 3.3.2.10

Combine the numerators over the common denominator.

$\frac{625 \cdot 5 - 32}{5}$

Step 3.3.2.11

Simplify the numerator.

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Step 3.3.2.11.1

Multiply $625$ by $5$ .

$\frac{3125 - 32}{5}$

Step 3.3.2.11.2

Subtract $32$ from $3125$ .

$\frac{3093}{5}$

Step 4

Simplify the root mean square formula.

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Step 4.1

Multiply $\frac{1}{5 - 2}$ by $\frac{3093}{5}$ .

$f_{rms} = \sqrt{\frac{3093}{(5 - 2) \cdot 5}}$

Step 4.2

Subtract $2$ from $5$ .

$f_{rms} = \sqrt{\frac{3093}{3 \cdot 5}}$

Step 4.3

Reduce the expression $\frac{3093}{3 \cdot 5}$ by cancelling the common factors.

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Step 4.3.1

Factor $3$ out of $3093$ .

$f_{rms} = \sqrt{\frac{3 \cdot 1031}{3 \cdot 5}}$

Step 4.3.2

Factor $3$ out of $3 \cdot 5$ .

$f_{rms} = \sqrt{\frac{3 \cdot 1031}{3 (5)}}$

Step 4.3.3

Cancel the common factor.

$f_{rms} = \sqrt{\frac{3 \cdot 1031}{3 \cdot 5}}$

Step 4.3.4

Rewrite the expression.

$f_{rms} = \sqrt{\frac{1031}{5}}$

Step 4.4

Rewrite $\sqrt{\frac{1031}{5}}$ as $\frac{\sqrt{1031}}{\sqrt{5}}$ .

$f_{rms} = \frac{\sqrt{1031}}{\sqrt{5}}$

Step 4.5

Multiply $\frac{\sqrt{1031}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$f_{rms} = \frac{\sqrt{1031}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 4.6

Combine and simplify the denominator.

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Step 4.6.1

Multiply $\frac{\sqrt{1031}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$f_{rms} = \frac{\sqrt{1031} \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 4.6.2

Raise $\sqrt{5}$ to the power of $1$ .

$f_{rms} = \frac{\sqrt{1031} \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 4.6.3

Raise $\sqrt{5}$ to the power of $1$ .

$f_{rms} = \frac{\sqrt{1031} \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 4.6.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f_{rms} = \frac{\sqrt{1031} \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 4.6.5

Add $1$ and $1$ .

$f_{rms} = \frac{\sqrt{1031} \sqrt{5}}{{\sqrt{5}}^{2}}$

Step 4.6.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 4.6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$f_{rms} = \frac{\sqrt{1031} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 4.6.6.2

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

$f_{rms} = \frac{\sqrt{1031} \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 4.6.6.3

Combine $\frac{1}{2}$ and $2$ .

$f_{rms} = \frac{\sqrt{1031} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 4.6.6.4

Cancel the common factor of $2$ .

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Step 4.6.6.4.1

Cancel the common factor.

$f_{rms} = \frac{\sqrt{1031} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 4.6.6.4.2

Rewrite the expression.

$f_{rms} = \frac{\sqrt{1031} \sqrt{5}}{5}$

Step 4.6.6.5

Evaluate the exponent.

$f_{rms} = \frac{\sqrt{1031} \sqrt{5}}{5}$

Step 4.7

Simplify the numerator.

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Step 4.7.1

Combine using the product rule for radicals.

$f_{rms} = \frac{\sqrt{1031 \cdot 5}}{5}$

Step 4.7.2

Multiply $1031$ by $5$ .

$f_{rms} = \frac{\sqrt{5155}}{5}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$f_{rms} = \frac{\sqrt{5155}}{5}$

Decimal Form:

$f_{rms} = 14.35966573 \dots$

Step 6

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