(a)

Step 1 :

The function is on the interval .

Average value of the function on is defined as .

Here .

Average value of is

\ \

.

Average value of the function on is .

Solution :

Average value of the function on is .

(b)

Step 1 :

The function is on the interval .

Average value of the function on is defined as .

Here .

Average value of is

Consider .

Differentiate with respect to .

Substitute and in equation (1).

Step 2 :

Power rule of integration : .

Substitute in above equation.

.

Average value of the function on is .

Solution :

Average value of the function on is .

(2)

Step 1 :

The function is on the interval .

Mean Value Theorem : .

Here .

Average value of is

.

Step 2 :

The function   in terms of is .

Substitute in .

and .

The value of is .

Solution :

The value of is .