# 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
Evaluate the integral.
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
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
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
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
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
Simplify the root mean square formula.
Step 4.1
Multiply by .
Step 4.2
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
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