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

Number of intervals .

Area is the sum of areas of the rectangles.

Width of the each rectangle : .

The rectangle repeat for and .

Lower estimates : 

Redraw the graph using left endpoints :

Area under the graph using left endpoints :

.

.

Observe the graph:

The value of the function at different values of are \ \

, , and .

Substitute , , ,and in .

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

Upper estimates :  \ \

Redraw the graph using right endpoints:

Area under the graph using right endpoints :

.

.

Observe the graph:

The value of the function at different values of are \ \

, , and .

Substitute , , , and in .

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

(b)

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

Observe the graph:

and .

Number of intervals .

Area is the sum of areas of the rectangles.

Width of the each rectangle: .

Determination of lower estimates:

.

.

.

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

Determination of upper estimates:

.

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

Lower estimate:

Graph of the function using left endpoint is

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

Upper estimate:

Graph of the function using right endpoint is

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

(b)

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

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