This may be cheating, but I used Excel to create a midpoint Riemann Sum with n=50.
That gives
Some of the numbers are skewed in the display, but that doesn't matter.
It turned out that way.
Code:
1 0.031416 0.000494
2 0.094248 0.004454
3 0.15708 0.012439
4 0.219912 0.024571
5 0.282743 0.041039
6 0.345575 0.062092
7 0.408407 0.088042
8 0.471239 0.119259
9 0.534071 0.156165
10 0.596903 0.199226
11 0.659735 0.248935
12 0.722566 0.305786
13 0.785398 0.37024
14 0.84823 0.442672
15 0.911062 0.5233
16 0.973894 0.612103
17 1.036726 0.70871
18 1.099558 0.812293
19 1.162389 0.921451
20 1.225221 1.034128
21 1.288053 1.147586
22 1.350885 1.258465
23 1.413717 1.362958
24 1.476549 1.457091
25 1.539381 1.537104
26 1.602212 1.599843
27 1.665044 1.643103
28 1.727876 1.665837
29 1.790708 1.668198
30 1.85354 1.651404
31 1.916372 1.617482
32 1.979204 1.568957
33 2.042035 1.508544
34 2.104867 1.438896
35 2.167699 1.362421
36 2.230531 1.281182
37 2.293363 1.196853
38 2.356195 1.11072
39 2.419027 1.023718
40 2.481858 0.936468
41 2.54469 0.849332
42 2.607522 0.762453
43 2.670354 0.675801
44 2.733186 0.589206
45 2.796018 0.502381
46 2.85885 0.414945
47 2.921681 0.326439
48 2.984513 0.236332
49 3.047345 0.144028
50 3.110177 0.04887
Add up the rightmost column and multiply by Pi/50 and you get
2.467659
Not too bad. If you run this through Maple or a TI-89 or something, you get
2.464010027.
Click on the links below to see the Riemann sum graphs of #1 and #2: