(a)

The function is  and .

Graph:

Graph the function  in the interval .

(b)

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

Number of sub intervals .

Area is the sum of areas of the  rectangles.

Width of the each rectangle :.

(i)

Determine area using right endpoints.

The rectangle repeat for  and .

Right end points are  and .

Area under the graph of the function using right endpoints :

.

Area of the graph of the function using right endpoints is .

Graph:

Graph the function  and rectangles using right endpoints:

(ii)

Mid points are  and .

Area under the graph of the function using mid point :

.

Area of the graph of the function using mid point is .

Graph:

Graph the function  and rectangles using mid point:

(c)

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

Number of subintervals .

Area is the sum of areas of the  rectangles.

Width of the each rectangle :.

The rectangle repeat for .

Right end points are .

Area under the graph of the function using right endpoints :

.

Area of the graph of the function using right endpoints is .

Graph :

Graph the function  and rectangles using right endpoints:

(ii)

Mid points are .

Area under the graph of the function using mid point :

. 

Area of the graph of the function using mid point is .

Graph:

Graph the function  and rectangles using mid point:

(a) Graph of the function  in the interval .

(b)

Area of the graph of the function using right endpoints is .

Graph of the function  and rectangles using right endpoints is

Area of the graph of the function using mid point is .

Graph the function  and rectangles using mid point is

(c)

Area of the graph of the function using right endpoints is .

Graph the function  and rectangles using right endpoints is

Area of the graph of the function using mid point is .

Graph the function  and rectangles using mid point is

.