Calculus Examples

$y = \sin (x)$ , $[- 2, 4]$

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.

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

Step 3

Evaluate the integral.

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Step 3.1

Use the half-angle formula to rewrite $\sin^{2} (x)$ as $\frac{1 - \cos (2 x)}{2}$ .

$\int_{- 2}^{4} \frac{1 - \cos (2 x)}{2} d x$

Step 3.2

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} \int_{- 2}^{4} 1 - \cos (2 x) d x$

Step 3.3

Split the single integral into multiple integrals.

$\frac{1}{2} (\int_{- 2}^{4} d x + \int_{- 2}^{4} - \cos (2 x) d x)$

Step 3.4

Apply the constant rule.

$\frac{1}{2} (x]_{- 2}^{4} + \int_{- 2}^{4} - \cos (2 x) d x)$

Step 3.5

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$\frac{1}{2} (x]_{- 2}^{4} - \int_{- 2}^{4} \cos (2 x) d x)$

Step 3.6

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

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Step 3.6.1

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

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Step 3.6.1.1

Differentiate $2 x$ .

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

Step 3.6.1.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

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

Step 3.6.1.3

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

$2 \cdot 1$

Step 3.6.1.4

Multiply $2$ by $1$ .

$2$

Step 3.6.2

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

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

Step 3.6.3

Multiply $2$ by $- 2$ .

$u_{lower} = - 4$

Step 3.6.4

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

$u_{upper} = 2 \cdot 4$

Step 3.6.5

Multiply $2$ by $4$ .

$u_{upper} = 8$

Step 3.6.6

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

$u_{lower} = - 4$

$u_{upper} = 8$

Step 3.6.7

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

$\frac{1}{2} (x]_{- 2}^{4} - \int_{- 4}^{8} \cos (u) \frac{1}{2} d u)$

Step 3.7

Combine $\cos (u)$ and $\frac{1}{2}$ .

$\frac{1}{2} (x]_{- 2}^{4} - \int_{- 4}^{8} \frac{\cos (u)}{2} d u)$

Step 3.8

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} (x]_{- 2}^{4} - (\frac{1}{2} \int_{- 4}^{8} \cos (u) d u))$

Step 3.9

The integral of $\cos (u)$ with respect to $u$ is $\sin (u)$ .

$\frac{1}{2} (x]_{- 2}^{4} - \frac{1}{2} \sin (u)]_{- 4}^{8})$

Step 3.10

Substitute and simplify.

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Step 3.10.1

Evaluate $x$ at $4$ and at $- 2$ .

$\frac{1}{2} ((4) + 2 - \frac{1}{2} \sin (u)]_{- 4}^{8})$

Step 3.10.2

Evaluate $\sin (u)$ at $8$ and at $- 4$ .

$\frac{1}{2} ((4) + 2 - \frac{1}{2} ((\sin (8)) - \sin (- 4)))$

Step 3.10.3

Add $4$ and $2$ .

$\frac{1}{2} (6 - \frac{1}{2} (\sin (8) - \sin (- 4)))$

Step 3.11

Simplify.

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Step 3.11.1

Simplify each term.

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Step 3.11.1.1

Simplify each term.

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Step 3.11.1.1.1

Evaluate $\sin (8)$ .

$\frac{1}{2} (6 - \frac{1}{2} (0.98935824 - \sin (- 4)))$

Step 3.11.1.1.2

Evaluate $\sin (- 4)$ .

$\frac{1}{2} (6 - \frac{1}{2} (0.98935824 - 1 \cdot 0.75680249))$

Step 3.11.1.1.3

Multiply $- 1$ by $0.75680249$ .

$\frac{1}{2} (6 - \frac{1}{2} (0.98935824 - 0.75680249))$

Step 3.11.1.2

Subtract $0.75680249$ from $0.98935824$ .

$\frac{1}{2} (6 - \frac{1}{2} \cdot 0.23255575)$

Step 3.11.1.3

Multiply $- \frac{1}{2} \cdot 0.23255575$ .

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Step 3.11.1.3.1

Multiply $0.23255575$ by $- 1$ .

$\frac{1}{2} (6 - 0.23255575 (\frac{1}{2}))$

Step 3.11.1.3.2

Combine $- 0.23255575$ and $\frac{1}{2}$ .

$\frac{1}{2} (6 + \frac{- 0.23255575}{2})$

Step 3.11.1.4

Divide $- 0.23255575$ by $2$ .

$\frac{1}{2} (6 - 0.11627787)$

Step 3.11.2

Subtract $0.11627787$ from $6$ .

$\frac{1}{2} \cdot 5.88372212$

Step 3.11.3

Combine $\frac{1}{2}$ and $5.88372212$ .

$\frac{5.88372212}{2}$

Step 3.11.4

Divide $5.88372212$ by $2$ .

$2.94186106$

Step 4

Simplify the root mean square formula.

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Step 4.1

Combine $\frac{1}{4 + 2}$ and $2.94186106$ .

$f_{rms} = \sqrt{\frac{2.94186106}{4 + 2}}$

Step 4.2

Simplify the expression.

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Step 4.2.1

Add $4$ and $2$ .

$f_{rms} = \sqrt{\frac{2.94186106}{6}}$

Step 4.2.2

Divide $2.94186106$ by $6$ .

$f_{rms} = \sqrt{0.49031017}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$f_{rms} = \sqrt{0.49031017}$

Decimal Form:

$f_{rms} = 0.70022151 \dots$