# 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

Since is constant with respect to , move out of the integral.

Step 6

By the Power Rule, the integral of with respect to is .

Step 7

Step 7.1

Combine and .

Step 7.2

Substitute and simplify.

Step 7.2.1

Evaluate at and at .

Step 7.2.2

Simplify.

Step 7.2.2.1

Raise to the power of .

Step 7.2.2.2

Cancel the common factor of and .

Step 7.2.2.2.1

Factor out of .

Step 7.2.2.2.2

Cancel the common factors.

Step 7.2.2.2.2.1

Factor out of .

Step 7.2.2.2.2.2

Cancel the common factor.

Step 7.2.2.2.2.3

Rewrite the expression.

Step 7.2.2.2.2.4

Divide by .

Step 7.2.2.3

Raising to any positive power yields .

Step 7.2.2.4

Cancel the common factor of and .

Step 7.2.2.4.1

Factor out of .

Step 7.2.2.4.2

Cancel the common factors.

Step 7.2.2.4.2.1

Factor out of .

Step 7.2.2.4.2.2

Cancel the common factor.

Step 7.2.2.4.2.3

Rewrite the expression.

Step 7.2.2.4.2.4

Divide by .

Step 7.2.2.5

Multiply by .

Step 7.2.2.6

Add and .

Step 7.2.2.7

Multiply by .

Step 8

Step 8.1

Multiply by .

Step 8.2

Add and .

Step 9

Step 9.1

Factor out of .

Step 9.2

Cancel the common factor.

Step 9.3

Rewrite the expression.

Step 10