The functions is  and the equations .

(a)

Graph of the lower sum:

Find lower sum.

The function  is increasing over the interval .

Number of subintervals are .

Width .

The left end point is .

.

The lower sum is .

The lower sum is  sq-units.

(b)

Graph of the upper sum:

Find upper sum.

The function  is increasing over the interval .

The right end point is .

.

Upper sum is .

The upper sum is  sq-units.

(c)

Graph of the sum using mid point rule:

Using mid point theorem:

The area is .

Consider .

Where ,  and .

Substitute  in .

.

Substitute  values in .

.

Using mid point theorem:

Area =

.

(d)

Find the formulas for lower sum, upper sum and middle point rule when number of subintervals are .

Find lower sum.

The function  is increasing over the interval .

The number of subintervals are .

The width .

The left end point is .

.

The lower sum is .

Find upper sum.

The right end point is .

Substitute  in .

.

Upper sum is .

.

Find the area by using mid point rule.

Using mid point theorem:

The area is .

Consider .

Where  is number of subintervals,  and .

Substitute  in .

.

Substitute  values in .

.

.

(e)

Complete the table by using,

 ,  and .

(f)

Since  is increasing function,  is always increasing and  is always decreasing.

(a) Graph of the lower sum:

 sq-units.

(b)

Graph of the upper sum:

 sq-units.

(c)

Graph of the sum using mid point rule:

 sq-units.

(d)

,   and .

(e)

(f)

Since  is increasing function,  is always increasing and  is always decreasing.