# Calculus Examples

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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.

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

Let . Then , so . Rewrite using and .

Combine fractions.

Write as a fraction with denominator .

Multiply and .

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

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

Simplify the answer.

Write as a fraction with denominator .

Multiply and .

Write as a fraction with denominator .

Multiply and .

Evaluate at and at .

Raise to the power of .

Raising to any positive power yields .

Reduce the expression by cancelling the common factors.

Factor out of .

Cancel the common factors.

Factor out of .

Cancel the common factor.

Rewrite the expression.

Divide by .

Multiply by .

Add and .

Rewrite as a product.

Multiply and .

Multiply by .

Reduce the expression by cancelling the common factors.

Factor out of .

Cancel the common factors.

Factor out of .

Cancel the common factor.

Rewrite the expression.

Simplify the expression.

Multiply by .

Multiply and .

Add and .

Reduce the expression by cancelling the common factors.

Factor out of .

Factor out of .

Cancel the common factor.

Rewrite the expression.

Rewrite as .

Simplify the numerator.

Rewrite as .

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

Multiply by .

Simplify.

Combine.

Raise to the power of .

Raise to the power of .

Use the power rule to combine exponents.

Add and .

Rewrite as .

The result can be shown in both exact and decimal forms.

Exact Form:

Decimal Form: