Estimate the area under the graph

Question

Suppose you estimate the area under the graph of f(x)=1x from x=11 to x=111 by adding the areas of rectangles as follows:
partition the interval into 50 equal subintervals and use the left endpoint of each interval to determine the height of the rectangle.
What is the area of the 26th rectangle?

Answer 1

Along the x-axis the interval from 11 to 111 is being divided into 20 equal subintervals. The length of each subinterval is (111 - 11) / 20 = 2.

 

If using the left endpoint of each subinterval to get the height/length of a typical rectangle consider:

 

1st rectangle left endpoint is: 11 + 0(2)

2nd                                 is: 11 + 1(2)

3rd                                  is: 11+ 2(2)

 

In other words a formula for the left endpoint of the nth rectangle is: endpoint = 11+ (n-1)d; this is just the formula for the nth term of an arithmetic sequence.

 

So, the 26th rectangle will have its left endpoint at 11 + (25)2 = 61. 

 

Since the length of the rectangle is the value of the function at that endpoint we have L = f(61) = 61, since f(x) = x.

 

Therefore the area of the 26th rectangle is A = LW = (61)(2) = 122 square units.

 

 

Answer 2

there is f(x)=x where x=11 to 111 then area equal to 111-11=100 sq.units. but divided into 50 equal intervals theneach rectangle is of 2sq.unit area i.e.100/50=2 which height of 1unit