# Calculus Examples

Find the Average Value of the Equation
,
Step 1
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:
Step 2
is continuous on .
is continuous
Step 3
The average value of function over the interval is defined as .
Step 4
Substitute the actual values into the formula for the average value of a function.
Step 5
Split the single integral into multiple integrals.
Step 6
Since is constant with respect to , move out of the integral.
Step 7
By the Power Rule, the integral of with respect to is .
Step 8
Combine and .
Step 9
Since is constant with respect to , move out of the integral.
Step 10
By the Power Rule, the integral of with respect to is .
Step 11
Combine and .
Substitute and simplify.
Evaluate at and at .
Evaluate at and at .
Simplify.
One to any power is one.
Raise to the power of .
Move the negative in front of the fraction.
Multiply by .
Multiply by .
Combine the numerators over the common denominator.
Cancel the common factor of and .
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by .
Multiply by .
One to any power is one.
Raise to the power of .
Combine the numerators over the common denominator.
Subtract from .
Cancel the common factor of and .
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by .
Multiply by .
Subtract from .
Step 12