For this first lab in Calculus 172, our purpose was to use the computing software Mathematica to
help us approximate an integral using Left Riemann Sums.
While many integrals can be computed by
using antiderivatives, some problems do not have elementary antiderivatives.
An example of this is
evaluating the integral of
e
x2
.
By using the program, we can generate a list of left end point sums which
will lead us to an approximate answer.
The idea is that, by increasing the amount of intervals used in the
sum, our accuracy will increase.
We started by determining the numerical value of the integral value of
e
x2
from 0 to .
Defining
π
this value as S, we get an answer of 0.886219.
Next, we used the program to estimate the value of S, if
we were to use the left end point method in subintervals numbering 2, 4, 8, 16, 32, 64, and 128.
The
results are shown in table 1 below.
2
0.09437560617670404
4
0.46624807552614694
8
0.11140388904891108
16 0.09006125003249066
32 0.09703177253605977
64 0.10414697878414458
128 0.1085715951239491
2
0.817788606
873612
4
0.392666366
30775764
8
0.196335537
1063762
16
0.098168670
36585408
32
0.049084586
62625237
64
0.024542357
959312255
128
0.012271195
255454614
Table 1
Table 2
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 Spring '08
 Pietraho
 Calculus, Numerical Analysis, Antiderivatives, Derivative, Integrals, Riemann Sums, left end, 16 0.098168670 36585408 32 0.049084586 62625237 64 0.024542357 959312255

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