# 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

Add and .

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

Add and .

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

Add and .

Step 13

Subtract from .

Step 14

Step 14.1

Factor out of .

Step 14.2

Cancel the common factor.

Step 14.3

Rewrite the expression.

Step 15