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

Division of the interval from x = 11 to x = 111 into 50 equal subintervals gives (111 − 11)/50 or 2 units

as the width of each subinterval.

Then the 26th rectangle will have its left-bottom endpoint at (x,y) = ([11+2×25],0) or (61,0). 

Since f(x) here is equal to 1x or x, the length of the 26th rectangle will be 61.

The area of the 26th rectangle is then approximately (2 × 61) or 122 square units.

Compare this estimated area to the exact area, which can be found by calculating

∫_{(from x = 61 to x = 63)}x dx which integrates to [x²/2|_{(from x = 61 to x = 63)}].

This last translates to {63² − 61²} ÷ 2 which gives exact area of the 26th rectangle as 124 square units.

The error of the estimate amounts to 1 − (122/124) equal to 0.01612903226 and equivalent to 1.6% error.