Calculus Examples

y = x - 2

$y = x - 2$ ,

(2, 7)

$(2, 7)$

Step 1

The Root Mean Square (RMS) of a function

f

$f$ over a specified interval

[a, b]

$[a, b]$ 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}

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

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)}

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

Step 3

Evaluate the integral.

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Step 3.1

Let

u = x - 2

$u = x - 2$ . Then

d u = d x

$d u = d x$ . Rewrite using

u

$u$ and

d

$d$

u

$u$ .

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Step 3.1.1

Let

u = x - 2

$u = x - 2$ . Find

\frac{d u}{d x}

$\frac{d u}{d x}$ .

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Step 3.1.1.1

Differentiate

x - 2

$x - 2$ .

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

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

Step 3.1.1.2

By the Sum Rule, the derivative of

x - 2

$x - 2$ with respect to

x

$x$ is

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

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

\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}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 1

$n = 1$ .

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

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

Step 3.1.1.4

Since

- 2

$- 2$ is constant with respect to

x

$x$ , the derivative of

- 2

$- 2$ with respect to

x

$x$ is

0

$0$ .

1 + 0

$1 + 0$

Step 3.1.1.5

Add

1

$1$ and

0

$0$ .

1

$1$

1

$1$

Step 3.1.2

Substitute the lower limit in for

x

$x$ in

u = x - 2

$u = x - 2$ .

u_{lower} = 2 - 2

$u_{lower} = 2 - 2$

Step 3.1.3

Subtract

2

$2$ from

2

$2$ .

u_{lower} = 0

$u_{lower} = 0$

Step 3.1.4

Substitute the upper limit in for

x

$x$ in

u = x - 2

$u = x - 2$ .

u_{upper} = 7 - 2

$u_{upper} = 7 - 2$

Step 3.1.5

Subtract

2

$2$ from

7

$7$ .

u_{upper} = 5

$u_{upper} = 5$

Step 3.1.6

The values found for

u_{lower}

$u_{lower}$ and

u_{upper}

$u_{upper}$ will be used to evaluate the definite integral.

u_{lower} = 0

$u_{lower} = 0$

u_{upper} = 5

$u_{upper} = 5$

Step 3.1.7

Rewrite the problem using

u

$u$ ,

d u

$d u$ , and the new limits of integration.

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

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

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

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

Step 3.2

By the Power Rule, the integral of

u^{2}

$u^{2}$ with respect to

u

$u$ is

\frac{1}{3} u^{3}

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

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

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

Step 3.3

Substitute and simplify.

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Step 3.3.1

Evaluate

\frac{1}{3} u^{3}

$\frac{1}{3} u^{3}$ at

5

$5$ and at

0

$0$ .

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

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

Step 3.3.2

Simplify.

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Step 3.3.2.1

Raise

5

$5$ to the power of

3

$3$ .

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

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

Step 3.3.2.2

Combine

\frac{1}{3}

$\frac{1}{3}$ and

125

$125$ .

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

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

Step 3.3.2.3

Raising

0

$0$ to any positive power yields

0

$0$ .

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

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

Step 3.3.2.4

Multiply

0

$0$ by

- 1

$- 1$ .

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

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

Step 3.3.2.5

Multiply

0

$0$ by

\frac{1}{3}

$\frac{1}{3}$ .

\frac{125}{3} + 0

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

Step 3.3.2.6

Add

\frac{125}{3}

$\frac{125}{3}$ and

0

$0$ .

\frac{125}{3}

$\frac{125}{3}$

\frac{125}{3}

$\frac{125}{3}$

\frac{125}{3}

$\frac{125}{3}$

\frac{125}{3}

$\frac{125}{3}$

Step 4

Simplify the root mean square formula.

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Step 4.1

Multiply

\frac{1}{7 - 2}

$\frac{1}{7 - 2}$ by

\frac{125}{3}

$\frac{125}{3}$ .

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

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

Step 4.2

Subtract

2

$2$ from

7

$7$ .

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

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

Step 4.3

Reduce the expression

\frac{125}{5 \cdot 3}

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

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Step 4.3.1

Factor

5

$5$ out of

125

$125$ .

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

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

Step 4.3.2

Factor

5

$5$ out of

5 \cdot 3

$5 \cdot 3$ .

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