# Calculus Examples

,

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

Apply the constant rule.

Step 10

Step 10.1

Evaluate at and at .

Step 10.2

Evaluate at and at .

Step 10.3

Simplify.

Step 10.3.1

Raise to the power of .

Step 10.3.2

Raising to any positive power yields .

Step 10.3.3

Cancel the common factor of and .

Step 10.3.3.1

Factor out of .

Step 10.3.3.2

Cancel the common factors.

Step 10.3.3.2.1

Factor out of .

Step 10.3.3.2.2

Cancel the common factor.

Step 10.3.3.2.3

Rewrite the expression.

Step 10.3.3.2.4

Divide by .

Step 10.3.4

Multiply by .

Step 10.3.5

Add and .

Step 10.3.6

Combine and .

Step 10.3.7

Multiply by .

Step 10.3.8

Cancel the common factor of and .

Step 10.3.8.1

Factor out of .

Step 10.3.8.2

Cancel the common factors.

Step 10.3.8.2.1

Factor out of .

Step 10.3.8.2.2

Cancel the common factor.

Step 10.3.8.2.3

Rewrite the expression.

Step 10.3.8.2.4

Divide by .

Step 10.3.9

Multiply by .

Step 10.3.10

Multiply by .

Step 10.3.11

Add and .

Step 10.3.12

Subtract from .

Step 11

Step 11.1

Multiply by .

Step 11.2

Add and .

Step 12

Step 12.1

Factor out of .

Step 12.2

Cancel the common factor.

Step 12.3

Rewrite the expression.

Step 13