The function  is increasing on the interval .

The interval  is divided into  subintervals.

(a)

Find the left end points.

The width .

Substitute  and .

.

The left end point is .

Substitute  and .

.

The left end point at the first subinterval.

Substitute  in .

.

The left end point at the last subinterval.

Substitute  in .

.

The left end points of the first and last subintervals are  and .

(b)

Find the right end points.

The left end point is .

Substitute  and .

.

The right end point at the first subinterval.

Substitute  in .

.

The right end point at the second subinterval.

Substitute  in .

.

The right end points of the first and last subintervals are  and .

(c)

When using the right end points on an increasing function the rectangles will above the graph of the function .

The upper right corner of each rectangle lies on .While the upper left corner of is above .

Consider  over the interval .

Graph the function:

Observe the graph:

In the right end points upper right corner of each rectangle lies on .

While the upper left corner of is above .

(d)

Consider a constant function  over interval .

Graph the function:

Observe the graph:

If a function is constant on the interval, then the height of rectangles are same.

(a) The left end points of the first and last subintervals are  and .

(b) The right end points of the first and last subintervals are  and .

(c)

In the right end points  upper right corner of each rectangle lies on .While the upper left corner of is above .

(d) If a function is constant on the interval, then the height of rectangles are same.