Page 271 - Calculus Demystified

CHAPTER 8
258 EXAMPLE 8.31 Applications of the Integral
1 −x 2
Use Simpson’s Rule to calculate 0 e dx to two decimal placesof
accuracy.
SOLUTION
If we set f(x) = e −x 2 then it is easy to calculate that
(iv) −x 2 2 4
f (x) = e ·[12 − 72x + 32x ].
Thus |f(x)|≤ 12 = M.
In order to achieve the desired degree of accuracy, we require that
M · (b − a) 5
180 · k 4 < 0.005
or
5
12 · 1 4 < 0.005.
180 · k 4LY
S =EAMF
Simple manipulation yields
200 <k .
15
This condition is satisfied when k = 2.
Thus our job is easy. We take the partition P ={0, 1/2, 1}. The sum arising
from Simpson’s Rule is then
1/2
1 T 3 {f(0) + 4f(1/2) + f(1)}
−0 2 −(1/2) 2 −1 2
= {e + 4 · e + e }
6
1
= {1 + 3.1152 + 0.3679}
6
1
≈ 6 · 4.4831
≈ 0.7472

Comparing with the “exact value” 0.746824 for the integral that we noted in
Example 8.29, we find that we have achieved two decimal places of accuracy.
It is interesting to note that if we had chosen a partition with k = 6, as we did
in Example 8.29, then Simpson’s Rule would have guaranteed an accuracy of

M · (b − a) 5 = 12 · 1 5 ≈ 0.00005144,
180 · k 4 180 · 6 4
or nearly four decimal places of accuracy.

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