(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

.