(a)

The function is .

Determine the area of the curve using three rectangles from  to .

Number of sub intervals .

Area is the sum of areas of the  rectangles.

Width of the each rectangle :.

The rectangle repeat for  and .

Right end points are  and .

Area under the graph using right endpoints :

.

Area under the graph using right endpoints is .

Graph :

Area under the graph using right endpoints :

Now, find the area for six rectangles.

Number of intervals .

Width of the each rectangle :.

The rectangle repeat for .

Right end points are .

Area under the graph using right endpoints :

.

Area under the graph using right endpoints is .

Graph :

Area under the graph using right endpoints :

(b)

Determine the area of the curve using three rectangles from  to .

Number of intervals .

Area is the sum of areas of the  rectangles.

Width of the each rectangle :.

The rectangle repeat for .

Left end points are .

Area under the graph using left endpoints : 

.

Area under the graph using left endpoints is .

Graph:

Area under the graph using left endpoints :

Now, find the area for six rectangles.

Number of intervals .

Width of the each rectangle :.

The rectangle repeat for .

Left end points are .

Area under the graph using right endpoints :

. 

Area under the graph using left endpoints is .

Graph :

Area under the graph using left endpoints :

(c)

Determine the area of the curve using mid points in three rectangles from  to .

Number of intervals .

Area is the sum of areas of the  rectangles.

Width of the each rectangle :.

Mid points are  and .

Area under the graph using mid points : 

.

Area under the graph using mid points is .

Graph :

Area under the graph using mid points :

Now, find the area for six rectangles.

Number of intervals .

Width of the each rectangle :.

Mid points are .

Area under the graph using mid points :

. 

Area under the graph using mid points is .

Graph :

Area under the graph using mid points :

(d)

The graph of the function is increasing function.

Right end points are under estimate of the function.

Left end points are overestimate of the function.

Result in part (c) are more accurate than (a) and (b), by observing the sketches. 

(a)  and .

(b)   and .

(c)

 and .

.