Calculus Examples

Find the Average Value of the Equation
,
The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.
Interval Notation:
Set-Builder Notation:
is continuous on .
is continuous
The average value of function over the interval is defined as .
Substitute the actual values into the formula for the average value of a function.
Split the single integral into multiple integrals.
Since is constant with respect to , move out of the integral.
By the Power Rule, the integral of with respect to is .
Combine and .
By the Power Rule, the integral of with respect to is .
Combine and .
Since is constant with respect to , move out of the integral.
Simplify the answer.
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Combine and .
Substitute and simplify.
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Evaluate at and at .
Evaluate at and at .
Raise to the power of .
Raise to the power of .
Combine the numerators over the common denominator.
Subtract from .
Reduce the expression by cancelling the common factors.
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Factor out of .
Cancel the common factors.
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by .
Multiply by .
Raise to the power of .
Multiply by .
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Combine.
Multiply by .
Combine the numerators over the common denominator.
Simplify the numerator.
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Multiply by .
Add and .
Raise to the power of .
Multiply by .
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Combine.
Multiply by .
Combine the numerators over the common denominator.
Simplify the numerator.
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Multiply by .
Add and .
Combine the numerators over the common denominator.
Subtract from .
Reduce the expression by cancelling the common factors.
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Factor out of .
Cancel the common factors.
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by .
Add and .
Subtract from .
Cancel the common factor of .
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Write as a fraction with denominator .
Factor out the greatest common factor .
Cancel the common factor.
Rewrite the expression.
Simplify.
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Multiply and .
Divide by .
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