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.
Evaluate the integral.
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Rewrite as .
Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Simplify the expression.
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Remove parentheses.
Reorder and .
Remove parentheses.
Move .
Multiply by to get .
Use the power rule to combine exponents.
Simplify by adding terms.
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Add and to get .
Simplify the expression.
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Add and to get .
Add and to get .
Multiply by to get .
Add and to get .
Subtract from to get .
Remove parentheses.
Since integration is linear, the integral of with respect to is .
By the Power Rule, the integral of with respect to is .
Combine fractions.
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Write as a fraction with denominator .
Multiply and to get .
Since is constant with respect to , the integral of with respect to is .
By the Power Rule, the integral of with respect to is .
Combine fractions.
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Write as a fraction with denominator .
Multiply and to get .
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.
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Write as a fraction with denominator .
Multiply and to get .
Evaluate at and at .
Evaluate at and at .
Evaluate at and at .
Simplify each term.
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Reduce the expression by cancelling the common factors.
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by to get .
Raise to the power of to get .
Raising to any positive power yields .
Divide by to get .
Multiply by to get .
Simplify each term.
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Raise to the power of to get .
Raising to any positive power yields .
Divide by to get .
Multiply by to get .
Add and to get .
Simplify .
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Write as a fraction with denominator .
Multiply and to get .
Multiply by to get .
Divide by to get .
Simplify each term.
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Raise to the power of to get .
Raising to any positive power yields .
Divide by to get .
Multiply by to get .
Add and to get .
Simplify .
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Write as a fraction with denominator .
Multiply and to get .
Multiply by to get .
Find the common denominator.
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Write as a fraction with denominator .
Multiply by .
Multiply and to get .
Write as a fraction with denominator .
Multiply by .
Multiply and to get .
Write as a fraction with denominator .
Multiply by .
Multiply and to get .
Combine the numerators over the common denominator.
Combine the numerators over the common denominator.
Combine the numerators over the common denominator.
Simplify the numerator.
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Multiply by to get .
Multiply by to get .
Multiply by to get .
Add and to get .
Subtract from to get .
Add and to get .
Simplify the root mean square formula.
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Simplify the expression.
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Multiply by to get .
Multiply and to get .
Simplify the expression.
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Multiply by to get .
Add and to get .
Reduce the expression by cancelling the common factors.
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Rewrite using the quotient rule for radicals.
Simplify the numerator.
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Rewrite as .
Pull terms out from under the radical.
Multiply by .
Simplify.
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Combine.
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Add and to get .
Rewrite as .
Simplify .
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Combine using the product rule for radicals.
Multiply by to get .
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