The function is  on interval is .

Number of rectangles are .

The sum of all inscribed rectangle is lower sum.

, where 

The sum of all circumscribed rectangle is upper sum.

, where 

Where .

Find lower sum.

.

Area of lower sum is .

Sum of natural numbers is .

The function is  on interval is .

Number of rectangles are .

Using mid point theorem:

The area is .

Consider .

.

Substitute in .

.

and .

Substitute  values.

.

.

.

.

.

Using mid point theorem:

Area =

sq-units.

Area of higher sum is 

Area of higher sum is .

.

Verification:

The function is and the interval .

Using mid point theorem:

The area is .

Consider . Take .

.

Substitute in .

.

.

Substitute  values from  to .

.

.

.

.

.

Using mid point theorem:

Area =

sq-units.

Graph the region: 

.

Graph the region when .

.

Observe the graph:

The approximate area when is .

Graph the region when .

.

Observe the graph:

The approximate area when is .

Graph the region when .

.

Observe the graph:

The approximate area when is .

Graph the region when .

.

Observe the graph:

The approximate area when is .

Complete the table:

Approximate area

Approximate area