Calculus Examples Applications of Integration

Find the Root Mean Square

$y = x - 2$ , $(2, 7)$

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.

$f_{rms} = \sqrt{\frac{1}{7 - 2} \cdot (\int_{2}^{7} {(x - 2)}^{2} d x)}$

Step 3

Evaluate the integral.

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Step 3.1

Let $u = x - 2$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 3.1.1

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

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Step 3.1.1.1

Differentiate $x - 2$ .

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

Step 3.1.1.2

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

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

Step 3.1.1.3

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

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

Step 3.1.1.4

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

$1 + 0$

Step 3.1.1.5

Add $1$ and $0$ .

$1$

Step 3.1.2

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

$u_{lower} = 2 - 2$

Step 3.1.3

Subtract $2$ from $2$ .

$u_{lower} = 0$

Step 3.1.4

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

$u_{upper} = 7 - 2$

Step 3.1.5

Subtract $2$ from $7$ .

$u_{upper} = 5$

Step 3.1.6

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

$u_{lower} = 0$

$u_{upper} = 5$

Step 3.1.7

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

$\int_{0}^{5} u^{2} d u$

Step 3.2

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

$\frac{1}{3} u^{3}]_{0}^{5}$

Step 3.3

Substitute and simplify.

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Step 3.3.1

Evaluate $\frac{1}{3} u^{3}$ at $5$ and at $0$ .

$(\frac{1}{3} \cdot 5^{3}) - \frac{1}{3} \cdot 0^{3}$

Step 3.3.2

Simplify.

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Step 3.3.2.1

Raise $5$ to the power of $3$ .

$\frac{1}{3} \cdot 125 - \frac{1}{3} \cdot 0^{3}$

Step 3.3.2.2

Combine $\frac{1}{3}$ and $125$ .

$\frac{125}{3} - \frac{1}{3} \cdot 0^{3}$

Step 3.3.2.3

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

$\frac{125}{3} - \frac{1}{3} \cdot 0$

Step 3.3.2.4

Multiply $0$ by $- 1$ .

$\frac{125}{3} + 0 (\frac{1}{3})$

Step 3.3.2.5

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

$\frac{125}{3} + 0$

Step 3.3.2.6

Add $\frac{125}{3}$ and $0$ .

$\frac{125}{3}$

Step 4

Simplify the root mean square formula.

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Step 4.1

Multiply $\frac{1}{7 - 2}$ by $\frac{125}{3}$ .

$f_{rms} = \sqrt{\frac{125}{(7 - 2) \cdot 3}}$

Step 4.2

Subtract $2$ from $7$ .

$f_{rms} = \sqrt{\frac{125}{5 \cdot 3}}$

Step 4.3

Reduce the expression $\frac{125}{5 \cdot 3}$ by cancelling the common factors.

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Step 4.3.1

Factor $5$ out of $125$ .

$f_{rms} = \sqrt{\frac{5 \cdot 25}{5 \cdot 3}}$

Step 4.3.2

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

$f_{rms} = \sqrt{\frac{5 \cdot 25}{5 (3)}}$

Step 4.3.3

Cancel the common factor.

$f_{rms} = \sqrt{\frac{5 \cdot 25}{5 \cdot 3}}$

Step 4.3.4

Rewrite the expression.

$f_{rms} = \sqrt{\frac{25}{3}}$

Step 4.4

Rewrite $\sqrt{\frac{25}{3}}$ as $\frac{\sqrt{25}}{\sqrt{3}}$ .

$f_{rms} = \frac{\sqrt{25}}{\sqrt{3}}$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Rewrite $25$ as $5^{2}$ .

$f_{rms} = \frac{\sqrt{5^{2}}}{\sqrt{3}}$

Step 4.5.2

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

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

Step 4.6

Multiply $\frac{5}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 4.7

Combine and simplify the denominator.

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Step 4.7.1

Multiply $\frac{5}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 4.7.2

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

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

Step 4.7.3

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

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

Step 4.7.4

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

$f_{rms} = \frac{5 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 4.7.5

Add $1$ and $1$ .

$f_{rms} = \frac{5 \sqrt{3}}{{\sqrt{3}}^{2}}$

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 $3^{\frac{1}{2}}$ .

$f_{rms} = \frac{5 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 4.7.6.2

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

$f_{rms} = \frac{5 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 4.7.6.3

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

$f_{rms} = \frac{5 \sqrt{3}}{3^{\frac{2}{2}}}$

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.

$f_{rms} = \frac{5 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 4.7.6.4.2

Rewrite the expression.

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

Step 4.7.6.5

Evaluate the exponent.

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$f_{rms} = 2.88675134 \dots$

Step 6

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Find the Root Mean Square

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