(2.a)

Step 1:

The function is  on interval is .

Number of rectangles are . 

The sum of all circumscribed rectangle is upper sum.

, where .

Width .

Find upper Sum.

Right end points : .

Area of higher sum is 

Step 2:

Apply summation formula  .

Apply summation formula  .

Apply summation formula .

The area of higher sum is .

Solution :

The area of the region is 2.37 sq-units.

(2.b)

Step 1:

The function is  on interval is .

Number of rectangles are .

The sum of all inscribed rectangle is lower sum.

, where .

Width .

Find lower Sum.

Left end points : .

Area of lower sum is .

Step 2:

Apply summation formula  .

Apply summation formula  .

Apply summation formula .

In this case .

The area of lower sum is .

Solution :

The area of the region is 10.37 sq-units.

(2.c)

Step 1:

The function is  on interval is .

Number of rectangles are .

Using mid point theorem:

The area is .

Consider .

Where , and .

Width .

.

Mid points :

Substitute i values from 1 to 4.

Step 2:

Using mid point theorem :

Area =

The area of the region is 6.815 sq-units.

Solution :

The area of the region is 6.815 sq-units.