# Calculus Examples

,

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

Add and .

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

Add and .

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

Raising to any positive power yields .

Step 3.5.2.4

Multiply by .

Step 3.5.2.5

Multiply by .

Step 3.5.2.6

Add and .

Step 3.5.2.7

Multiply by .

Step 3.5.2.8

Multiply by .

Step 3.5.2.9

Cancel the common factor of and .

Step 3.5.2.9.1

Factor out of .

Step 3.5.2.9.2

Cancel the common factors.

Step 3.5.2.9.2.1

Factor out of .

Step 3.5.2.9.2.2

Cancel the common factor.

Step 3.5.2.9.2.3

Rewrite the expression.

Step 4

Step 4.1

Multiply by .

Step 4.2

Add and .

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 4.5.2

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

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 5

The result can be shown in multiple forms.

Exact Form:

Decimal Form:

Step 6