# Calculus Examples

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Step 1

The Root Mean Square (RMS) of a function over a specified interval is the square root of the arithmetic mean (average) of the squares of the original values.

Step 2

Substitute the actual values into the formula for the root mean square of a function.

Step 3

Step 3.1

Let . Then , so . Rewrite using and .

Step 3.1.1

Let . Find .

Step 3.1.1.1

Differentiate .

Step 3.1.1.2

By the Sum Rule, the derivative of with respect to is .

Step 3.1.1.3

Evaluate .

Step 3.1.1.3.1

Since is constant with respect to , the derivative of with respect to is .

Step 3.1.1.3.2

Differentiate using the Power Rule which states that is where .

Step 3.1.1.3.3

Multiply by .

Step 3.1.1.4

Differentiate using the Constant Rule.

Step 3.1.1.4.1

Since is constant with respect to , the derivative of with respect to is .

Step 3.1.1.4.2

Add and .

Step 3.1.2

Substitute the lower limit in for in .

Step 3.1.3

Simplify.

Step 3.1.3.1

Multiply by .

Step 3.1.3.2

Subtract from .

Step 3.1.4

Substitute the upper limit in for in .

Step 3.1.5

Simplify.

Step 3.1.5.1

Multiply by .

Step 3.1.5.2

Subtract from .

Step 3.1.6

The values found for and will be used to evaluate the definite integral.

Step 3.1.7

Rewrite the problem using , , and the new limits of integration.

Step 3.2

Combine and .

Step 3.3

Since is constant with respect to , move out of the integral.

Step 3.4

By the Power Rule, the integral of with respect to is .

Step 3.5

Substitute and simplify.

Step 3.5.1

Evaluate at and at .

Step 3.5.2

Simplify.

Step 3.5.2.1

Raise to the power of .

Step 3.5.2.2

Combine and .

Step 3.5.2.3

Raise to the power of .

Step 3.5.2.4

Multiply by .

Step 3.5.2.5

Combine and .

Step 3.5.2.6

Move the negative in front of the fraction.

Step 3.5.2.7

Combine the numerators over the common denominator.

Step 3.5.2.8

Subtract from .

Step 3.5.2.9

Multiply by .

Step 3.5.2.10

Multiply by .

Step 3.5.2.11

Cancel the common factor of and .

Step 3.5.2.11.1

Factor out of .

Step 3.5.2.11.2

Cancel the common factors.

Step 3.5.2.11.2.1

Factor out of .

Step 3.5.2.11.2.2

Cancel the common factor.

Step 3.5.2.11.2.3

Rewrite the expression.

Step 4

Step 4.1

Multiply by .

Step 4.2

Subtract from .

Step 4.3

Reduce the expression by cancelling the common factors.

Step 4.3.1

Factor out of .

Step 4.3.2

Factor out of .

Step 4.3.3

Cancel the common factor.

Step 4.3.4

Rewrite the expression.

Step 4.4

Rewrite as .

Step 4.5

Simplify the numerator.

Step 4.5.1

Rewrite as .

Step 4.5.1.1

Factor out of .

Step 4.5.1.2

Rewrite as .

Step 4.5.2

Pull terms out from under the radical.

Step 4.6

Multiply by .

Step 4.7

Combine and simplify the denominator.

Step 4.7.1

Multiply by .

Step 4.7.2

Raise to the power of .

Step 4.7.3

Raise to the power of .

Step 4.7.4

Use the power rule to combine exponents.

Step 4.7.5

Add and .

Step 4.7.6

Rewrite as .

Step 4.7.6.1

Use to rewrite as .

Step 4.7.6.2

Apply the power rule and multiply exponents, .

Step 4.7.6.3

Combine and .

Step 4.7.6.4

Cancel the common factor of .

Step 4.7.6.4.1

Cancel the common factor.

Step 4.7.6.4.2

Rewrite the expression.

Step 4.7.6.5

Evaluate the exponent.

Step 4.8

Simplify the numerator.

Step 4.8.1

Combine using the product rule for radicals.

Step 4.8.2

Multiply by .

Step 5

The result can be shown in multiple forms.

Exact Form:

Decimal Form:

Step 6