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PAGE: 303SET: ExercisesPROBLEM: 2
Please look in your text book for this problem Statement

Determine the area of the curve using six 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 .

Left end points are  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:

Right end points are  and .

Area under the graph using right endpoints :

.

.

Observe the graph:

The value of the function at different values of  are

  and .

Substituteandin .

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

(iii)

Mid point estimates : 

Redraw the graph using midpoints:

Mid points are  and .

Mid point estimation is

 .

Observe the graph:

The value of the function at different values of  are

 and .

Substitute andin .

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

(b)

The graph of the function is decreasing function.

From the  graph of lower estimate , it is observed that the bars are above the graph of the curve.

Therefore, is an overestimate of the area.

(c)

The graph of the function is decreasing function.

From the  graph of left end estimate , it is observed that the bars are below the graph of the curve.

Therefore,  is an underestimate of the area.

(d)

The function is decreasing function.

Since the  is an overestimate of the area and  is an overestimate of the area.

Hence we consider  as the best estimation.

Area under the graph using left endpoints is

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

Area under the graph using right endpoints is

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

Area under the graph using Mid points is

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

(b)  is an overestimate of the true area.

(c)  is an  underestimate of the true area.

(d)  is the best estimation.



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