# Calculus Examples

,
Step 1
Write as a function.
Step 2
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 3
is continuous on .
is continuous
Step 4
The average value of function over the interval is defined as .
Step 5
Substitute the actual values into the formula for the average value of a function.
Step 6
Split the single integral into multiple integrals.
Step 7
Since is constant with respect to , move out of the integral.
Step 8
By the Power Rule, the integral of with respect to is .
Step 9
Combine and .
Step 10
By the Power Rule, the integral of with respect to is .
Step 11
Apply the constant rule.
Step 12
Step 12.1
Combine and .
Step 12.2
Substitute and simplify.
Step 12.2.1
Evaluate at and at .
Step 12.2.2
Evaluate at and at .
Step 12.2.3
Simplify.
Step 12.2.3.1
Raise to the power of .
Step 12.2.3.2
Raise to the power of .
Step 12.2.3.3
Combine the numerators over the common denominator.
Step 12.2.3.4
Subtract from .
Step 12.2.3.5
Cancel the common factor of and .
Step 12.2.3.5.1
Factor out of .
Step 12.2.3.5.2
Cancel the common factors.
Step 12.2.3.5.2.1
Factor out of .
Step 12.2.3.5.2.2
Cancel the common factor.
Step 12.2.3.5.2.3
Rewrite the expression.
Step 12.2.3.5.2.4
Divide by .
Step 12.2.3.6
Multiply by .
Step 12.2.3.7
Raise to the power of .
Step 12.2.3.8
Combine and .
Step 12.2.3.9
Multiply by .
Step 12.2.3.10
To write as a fraction with a common denominator, multiply by .
Step 12.2.3.11
Combine and .
Step 12.2.3.12
Combine the numerators over the common denominator.
Step 12.2.3.13
Simplify the numerator.
Step 12.2.3.13.1
Multiply by .
Step 12.2.3.13.2
Step 12.2.3.14
Raise to the power of .
Step 12.2.3.15
Combine and .
Step 12.2.3.16
Multiply by .
Step 12.2.3.17
To write as a fraction with a common denominator, multiply by .
Step 12.2.3.18
Combine and .
Step 12.2.3.19
Combine the numerators over the common denominator.
Step 12.2.3.20
Simplify the numerator.
Step 12.2.3.20.1
Multiply by .
Step 12.2.3.20.2
Step 12.2.3.21
Combine the numerators over the common denominator.
Step 12.2.3.22
Subtract from .
Step 12.2.3.23
Cancel the common factor of and .
Step 12.2.3.23.1
Factor out of .
Step 12.2.3.23.2
Cancel the common factors.
Step 12.2.3.23.2.1
Factor out of .
Step 12.2.3.23.2.2
Cancel the common factor.
Step 12.2.3.23.2.3
Rewrite the expression.
Step 12.2.3.23.2.4
Divide by .
Step 12.2.3.24
Step 13
Subtract from .
Step 14
Cancel the common factor of .
Step 14.1
Factor out of .
Step 14.2
Cancel the common factor.
Step 14.3
Rewrite the expression.
Step 15