Refining a subdivision

Better Approximation to Areas by Refinement

To approximate the area between the graph of a function, say f(x) = 1/(x⁴+1), and the x-axis and between two x-values, say x=0 and x=2, we found a "Riemann sum", the sum of areas of rectangles with bases the subintervals determined by a subdivision 0=x₀ < x₁ < x₂ < ... < x_n=2 of the interval from 0 to 2, and where the height of the i-th rectangle is the value of f at a point x*_i in the i-th subinterval. Now we want to see that if we "refine" the subdivision, adding more points and shortening the length of the largest subinterval in the subdivision, we will improve the approximation to the area.

We begin by comparing the (maximum) Riemann sums for this function on this interval with 2, 4, 10 and 20 equal subintervals.

First we compare the sums for 2 and 4 subintervals. We can see that the sum for 4 subintervals improves the approximation to the area by the area of the region in yellow.

Next we compare the sums for 2 and 10 subintervals. The latter approximation is closer to the area under the curve by the area of the gray region.

Finally (in this example) we compare the sums for 10 and 20 subintervals. The approximation with 20 is closer to the area under the curve than the one with 10 by the area of the violet region. Note also that, since the subdivision into 20 is also a refinement of the subdivision into 4 subintervals, the former is also a better approximation that the latter. On the other hand, since the subdivision into 10 subintervals is not a refinement of the subdivision into 4, there is no obvious way to see that 10 gives a better approximation than 4 (although it certainly appears to do so in this case).

The graphic to the left shows dynamically how increasing the number n of subintervals gives better approximations to the area under the curve. It moves quickly through all the values of n from 1 to 10, then slows a bit as the value of n doubles three times to 80. For each n-value, the subintervals are all of equal length and the maximum Riemann sum is shown. Though it is not clear (and may not be true) that each approximation is better than the last, it is clear that the approximation is very close for the larger values of n.

To make the same point numerically rather than graphically, we use a table taken from a spreadsheet, showing approximations to the area using 10, 20 and 40 equal subintervals. The length of the column of 40 values is a bit daunting, but the important information is collected at the top (in all cases): The top row shows the maximum and minimum Riemann sum approximations, and the second is the differences between them. Note that, as the number of subintervals is doubled, the difference between the maximum and minimum Riemann sum is roughly (in fact, in this case, apparently exactly) halved. Since the exact area is between the maximum and minimum Riemann sums, the finer subdivisions give better approximations to the desired area.

10 Max Min 20 Max Min 40 Max Min

Rsums 1.163878 0.975642 1.117094 1.022977 1.093634 1.046575

Diffs 0.188235 0.094118 0.047059

x_i f(x_i-1) f(x_i) x_i f(x_i-1) f(x_i) x_i f(x_i-1) f(x_i)
0 1 0.998403 0 1 0.9999 0 1 0.999994

0.2 0.998403 0.975039 0.1 0.9999 0.998403 0.05 0.999994 0.9999

0.4 0.975039 0.885269 0.2 0.998403 0.991965 0.1 0.9999 0.999494

0.6 0.885269 0.709421 0.3 0.991965 0.975039 0.15 0.999494 0.998403

0.8 0.709421 0.5 0.4 0.975039 0.941176 0.2 0.998403 0.996109

1 0.5 0.325351 0.5 0.941176 0.885269 0.25 0.996109 0.991965

1.2 0.325351 0.206543 0.6 0.885269 0.806387 0.3 0.991965 0.985216

1.4 0.206543 0.132387 0.7 0.806387 0.709421 0.35 0.985216 0.975039

1.6 0.132387 0.086975 0.8 0.709421 0.603828 0.4 0.975039 0.960609

1.8 0.086975 0.058824 0.9 0.603828 0.5 0.45 0.960609 0.941176

2 1 0.5 0.405828 0.5 0.941176 0.916165

1.1 0.405828 0.325351 0.55 0.916165 0.885269

1.2 0.325351 0.259329 0.6 0.885269 0.848532

1.3 0.259329 0.206543 0.65 0.848532 0.806387

1.4 0.206543 0.164948 0.7 0.806387 0.759644

1.5 0.164948 0.132387 0.75 0.759644 0.709421

1.6 0.132387 0.106928 0.8 0.709421 0.657028

1.7 0.106928 0.086975 0.85 0.657028 0.603828

1.8 0.086975 0.071265 0.9 0.603828 0.551114

1.9 0.071265 0.058824 0.95 0.551114 0.5

2 1 0.5 0.451364

1.05 0.451364 0.405828

1.1 0.405828 0.363768

1.15 0.363768 0.325351

1.2 0.325351 0.290579

1.25 0.290579 0.259329

1.3 0.259329 0.231401

1.35 0.231401 0.206543

1.4 0.206543 0.184485

1.45 0.184485 0.164948

1.5 0.164948 0.147667

1.55 0.147667 0.132387

1.6 0.132387 0.118878

1.65 0.118878 0.106928

1.7 0.106928 0.096349

1.75 0.096349 0.086975

1.8 0.086975 0.078657

1.85 0.078657 0.071265

1.9 0.071265 0.064687

1.95 0.064687 0.058824

2

	First we compare the sums for 2 and 4 subintervals. We can see that the sum for 4 subintervals improves the approximation to the area by the area of the region in yellow.
	Next we compare the sums for 2 and 10 subintervals. The latter approximation is closer to the area under the curve by the area of the gray region.
	Finally (in this example) we compare the sums for 10 and 20 subintervals. The approximation with 20 is closer to the area under the curve than the one with 10 by the area of the violet region. Note also that, since the subdivision into 20 is also a refinement of the subdivision into 4 subintervals, the former is also a better approximation that the latter. On the other hand, since the subdivision into 10 subintervals is not a refinement of the subdivision into 4, there is no obvious way to see that 10 gives a better approximation than 4 (although it certainly appears to do so in this case).
	The graphic to the left shows dynamically how increasing the number n of subintervals gives better approximations to the area under the curve. It moves quickly through all the values of n from 1 to 10, then slows a bit as the value of n doubles three times to 80. For each n-value, the subintervals are all of equal length and the maximum Riemann sum is shown. Though it is not clear (and may not be true) that each approximation is better than the last, it is clear that the approximation is very close for the larger values of n.

10	Max	Min	20	Max	Min	40	Max	Min
Rsums	1.163878	0.975642		1.117094	1.022977		1.093634	1.046575
Diffs		0.188235			0.094118			0.047059
x_i	f(x_i-1)	f(x_i)	x_i	f(x_i-1)	f(x_i)	x_i	f(x_i-1)	f(x_i)
0	1	0.998403	0	1	0.9999	0	1	0.999994
0.2	0.998403	0.975039	0.1	0.9999	0.998403	0.05	0.999994	0.9999
0.4	0.975039	0.885269	0.2	0.998403	0.991965	0.1	0.9999	0.999494
0.6	0.885269	0.709421	0.3	0.991965	0.975039	0.15	0.999494	0.998403
0.8	0.709421	0.5	0.4	0.975039	0.941176	0.2	0.998403	0.996109
1	0.5	0.325351	0.5	0.941176	0.885269	0.25	0.996109	0.991965
1.2	0.325351	0.206543	0.6	0.885269	0.806387	0.3	0.991965	0.985216
1.4	0.206543	0.132387	0.7	0.806387	0.709421	0.35	0.985216	0.975039
1.6	0.132387	0.086975	0.8	0.709421	0.603828	0.4	0.975039	0.960609
1.8	0.086975	0.058824	0.9	0.603828	0.5	0.45	0.960609	0.941176
2			1	0.5	0.405828	0.5	0.941176	0.916165
			1.1	0.405828	0.325351	0.55	0.916165	0.885269
			1.2	0.325351	0.259329	0.6	0.885269	0.848532
			1.3	0.259329	0.206543	0.65	0.848532	0.806387
			1.4	0.206543	0.164948	0.7	0.806387	0.759644
			1.5	0.164948	0.132387	0.75	0.759644	0.709421
			1.6	0.132387	0.106928	0.8	0.709421	0.657028
			1.7	0.106928	0.086975	0.85	0.657028	0.603828
			1.8	0.086975	0.071265	0.9	0.603828	0.551114
			1.9	0.071265	0.058824	0.95	0.551114	0.5
			2			1	0.5	0.451364
						1.05	0.451364	0.405828
						1.1	0.405828	0.363768
						1.15	0.363768	0.325351
						1.2	0.325351	0.290579
						1.25	0.290579	0.259329
						1.3	0.259329	0.231401
						1.35	0.231401	0.206543
						1.4	0.206543	0.184485
						1.45	0.184485	0.164948
						1.5	0.164948	0.147667
						1.55	0.147667	0.132387
						1.6	0.132387	0.118878
						1.65	0.118878	0.106928
						1.7	0.106928	0.096349
						1.75	0.096349	0.086975
						1.8	0.086975	0.078657
						1.85	0.078657	0.071265
						1.9	0.071265	0.064687
						1.95	0.064687	0.058824
						2