# 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 . Rewrite using and .

Let . Find .

Differentiate .

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

Differentiate using the Power Rule which states that is where .

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

Add and .

Substitute the lower limit in for in .

Subtract from .

Substitute the upper limit in for in .

Subtract from .

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

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

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

Substitute and simplify.

Evaluate at and at .

Simplify.

Raise to the power of .

Combine and .

Raising to any positive power yields .

Multiply by .

Multiply by .

Add and .

Multiply and .

Subtract from .

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 .

Combine and simplify the denominator.

Multiply and .

Raise to the power of .

Raise to the power of .

Use the power rule to combine exponents.

Add and .

Rewrite as .

Rewrite as .

Apply the power rule and multiply exponents, .

Combine and .

Cancel the common factor of .

Cancel the common factor.

Divide by .

Evaluate the exponent.

The result can be shown in multiple forms.

Exact Form:

Decimal Form: