Calculus Examples

$y = 4 x - 2$ , $(1, 3)$

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.

$f_{rms} = \sqrt{\frac{1}{3 - 1} \cdot (\int_{1}^{3} {(4 x - 2)}^{2} d x)}$

Step 3

Evaluate the integral.

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Step 3.1

Let $u = 4 x - 2$ . Then $d u = 4 d x$ , so $\frac{1}{4} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 3.1.1

Let $u = 4 x - 2$ . Find $\frac{d u}{d x}$ .

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Step 3.1.1.1

Differentiate $4 x - 2$ .

$\frac{d}{d x} [4 x - 2]$

Step 3.1.1.2

By the Sum Rule, the derivative of $4 x - 2$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [- 2]$ .

$\frac{d}{d x} [4 x] + \frac{d}{d x} [- 2]$

Step 3.1.1.3

Evaluate $\frac{d}{d x} [4 x]$ .

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Step 3.1.1.3.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$4 \frac{d}{d x} [x] + \frac{d}{d x} [- 2]$

Step 3.1.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$4 \cdot 1 + \frac{d}{d x} [- 2]$

Step 3.1.1.3.3

Multiply $4$ by $1$ .

$4 + \frac{d}{d x} [- 2]$

Step 3.1.1.4

Differentiate using the Constant Rule.

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Step 3.1.1.4.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$4 + 0$

Step 3.1.1.4.2

Add $4$ and $0$ .

$4$

Step 3.1.2

Substitute the lower limit in for $x$ in $u = 4 x - 2$ .

$u_{lower} = 4 \cdot 1 - 2$

Step 3.1.3

Simplify.

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Step 3.1.3.1

Multiply $4$ by $1$ .

$u_{lower} = 4 - 2$

Step 3.1.3.2

Subtract $2$ from $4$ .

$u_{lower} = 2$

Step 3.1.4

Substitute the upper limit in for $x$ in $u = 4 x - 2$ .

$u_{upper} = 4 \cdot 3 - 2$

Step 3.1.5

Simplify.

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Step 3.1.5.1

Multiply $4$ by $3$ .

$u_{upper} = 12 - 2$

Step 3.1.5.2

Subtract $2$ from $12$ .

$u_{upper} = 10$

Step 3.1.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 2$

$u_{upper} = 10$

Step 3.1.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$\int_{2}^{10} u^{2} \frac{1}{4} d u$

Step 3.2

Combine $u^{2}$ and $\frac{1}{4}$ .

$\int_{2}^{10} \frac{u^{2}}{4} d u$

Step 3.3

Since $\frac{1}{4}$ is constant with respect to $u$ , move $\frac{1}{4}$ out of the integral.

$\frac{1}{4} \int_{2}^{10} u^{2} d u$

Step 3.4

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

$\frac{1}{4} \frac{1}{3} u^{3}]_{2}^{10}$

Step 3.5

Substitute and simplify.

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Step 3.5.1

Evaluate $\frac{1}{3} u^{3}$ at $10$ and at $2$ .

$\frac{1}{4} ((\frac{1}{3} \cdot 10^{3}) - \frac{1}{3} \cdot 2^{3})$

Step 3.5.2

Simplify.

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Step 3.5.2.1

Raise $10$ to the power of $3$ .

$\frac{1}{4} (\frac{1}{3} \cdot 1000 - \frac{1}{3} \cdot 2^{3})$

Step 3.5.2.2

Combine $\frac{1}{3}$ and $1000$ .

$\frac{1}{4} (\frac{1000}{3} - \frac{1}{3} \cdot 2^{3})$

Step 3.5.2.3

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

$\frac{1}{4} (\frac{1000}{3} - \frac{1}{3} \cdot 8)$

Step 3.5.2.4

Multiply $8$ by $- 1$ .

$\frac{1}{4} (\frac{1000}{3} - 8 (\frac{1}{3}))$

Step 3.5.2.5

Combine $- 8$ and $\frac{1}{3}$ .

$\frac{1}{4} (\frac{1000}{3} + \frac{- 8}{3})$

Step 3.5.2.6

Move the negative in front of the fraction.

$\frac{1}{4} (\frac{1000}{3} - \frac{8}{3})$

Step 3.5.2.7

Combine the numerators over the common denominator.

$\frac{1}{4} \cdot \frac{1000 - 8}{3}$

Step 3.5.2.8

Subtract $8$ from $1000$ .

$\frac{1}{4} \cdot \frac{992}{3}$

Step 3.5.2.9

Multiply $\frac{1}{4}$ by $\frac{992}{3}$ .

$\frac{992}{4 \cdot 3}$

Step 3.5.2.10

Multiply $4$ by $3$ .

$\frac{992}{12}$

Step 3.5.2.11

Cancel the common factor of $992$ and $12$ .

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Step 3.5.2.11.1

Factor $4$ out of $992$ .

$\frac{4 (248)}{12}$

Step 3.5.2.11.2

Cancel the common factors.

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Step 3.5.2.11.2.1

Factor $4$ out of $12$ .

$\frac{4 \cdot 248}{4 \cdot 3}$

Step 3.5.2.11.2.2

Cancel the common factor.

$\frac{4 \cdot 248}{4 \cdot 3}$

Step 3.5.2.11.2.3

Rewrite the expression.

$\frac{248}{3}$

Step 4

Simplify the root mean square formula.

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Step 4.1

Multiply $\frac{1}{3 - 1}$ by $\frac{248}{3}$ .

$f_{rms} = \sqrt{\frac{248}{(3 - 1) \cdot 3}}$

Step 4.2

Subtract $1$ from $3$ .

$f_{rms} = \sqrt{\frac{248}{2 \cdot 3}}$

Step 4.3

Reduce the expression $\frac{248}{2 \cdot 3}$ by cancelling the common factors.

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Step 4.3.1

Factor $2$ out of $248$ .

$f_{rms} = \sqrt{\frac{2 \cdot 124}{2 \cdot 3}}$

Step 4.3.2

Factor $2$ out of $2 \cdot 3$ .

$f_{rms} = \sqrt{\frac{2 \cdot 124}{2 (3)}}$

Step 4.3.3

Cancel the common factor.

$f_{rms} = \sqrt{\frac{2 \cdot 124}{2 \cdot 3}}$

Step 4.3.4

Rewrite the expression.

$f_{rms} = \sqrt{\frac{124}{3}}$

Step 4.4

Rewrite $\sqrt{\frac{124}{3}}$ as $\frac{\sqrt{124}}{\sqrt{3}}$ .

$f_{rms} = \frac{\sqrt{124}}{\sqrt{3}}$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Rewrite $124$ as $2^{2} \cdot 31$ .

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Step 4.5.1.1

Factor $4$ out of $124$ .

$f_{rms} = \frac{\sqrt{4 (31)}}{\sqrt{3}}$

Step 4.5.1.2

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

$f_{rms} = \frac{\sqrt{2^{2} \cdot 31}}{\sqrt{3}}$

Step 4.5.2

Pull terms out from under the radical.

$f_{rms} = \frac{2 \sqrt{31}}{\sqrt{3}}$

Step 4.6

Multiply $\frac{2 \sqrt{31}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$f_{rms} = \frac{2 \sqrt{31}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 4.7

Combine and simplify the denominator.

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Step 4.7.1

Multiply $\frac{2 \sqrt{31}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$f_{rms} = \frac{2 \sqrt{31} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 4.7.2

Raise $\sqrt{3}$ to the power of $1$ .

$f_{rms} = \frac{2 \sqrt{31} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 4.7.3

Raise $\sqrt{3}$ to the power of $1$ .

$f_{rms} = \frac{2 \sqrt{31} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 4.7.4

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

$f_{rms} = \frac{2 \sqrt{31} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 4.7.5

Add $1$ and $1$ .

$f_{rms} = \frac{2 \sqrt{31} \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 4.7.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 4.7.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$f_{rms} = \frac{2 \sqrt{31} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 4.7.6.2

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

$f_{rms} = \frac{2 \sqrt{31} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 4.7.6.3

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

$f_{rms} = \frac{2 \sqrt{31} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 4.7.6.4

Cancel the common factor of $2$ .

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Step 4.7.6.4.1

Cancel the common factor.

$f_{rms} = \frac{2 \sqrt{31} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 4.7.6.4.2

Rewrite the expression.

$f_{rms} = \frac{2 \sqrt{31} \sqrt{3}}{3}$

Step 4.7.6.5

Evaluate the exponent.

$f_{rms} = \frac{2 \sqrt{31} \sqrt{3}}{3}$

Step 4.8

Simplify the numerator.

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Step 4.8.1

Combine using the product rule for radicals.

$f_{rms} = \frac{2 \sqrt{3 \cdot 31}}{3}$

Step 4.8.2

Multiply $3$ by $31$ .

$f_{rms} = \frac{2 \sqrt{93}}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$f_{rms} = \frac{2 \sqrt{93}}{3}$

Decimal Form:

$f_{rms} = 6.42910050 \dots$

Step 6

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