Page 272 - Calculus Demystified

CHAPTER 8
Applications of the Integral
EXAMPLE 8.32 259
Estimate the integral
1
1
dx
1 + x 2
0
using Simpson’s Rule with a partition having four intervals. What degree of
accuracy doesthisrepresent?

SOLUTION
Of course this example is parallel to Example 8.30, and you should compare
2
the two examples. Our function is f(x) = 1/(1 + x ) and our partition is
P ={0, 1/4, 2/4, 3/4, 1}. The sum from Simpson’s Rule is

1/4
S = ·{f(0) + 4f(1/4) + 2f(1/2) + 4f(3/4) + f(1)}
3
)
1 1 1
= · + 4 ·
12 1 + 0 2 1 + (1/4) 2
*
1 1 1
+ 2 · + 4 · +
1 + (1/2) 2 1 + (3/4) 2 1 + 1 2
1
≈ ·{1 + 3.7647 + 1.6 + 2.56 + 0.5}
12
≈ 0.785392.

Comparing with Example 8.30, we see that this answer is accurate to four
decimal places. We invite the reader to do the necessary calculation with the
Simpson’s Rule error to term to confirm that we could have predicted this
degree of accuracy.

You Try It: Estimate the integral

e 1
2
dx
ln x
e
using both the Trapezoid Rule and Simpson’s Rule with a partition having six
points. Use the error term estimate to state what the accuracy prediction of each of
your calculations is. If the software Mathematica or Maple is available to you,
check the answers you have obtained against those provided by these computer
algebra systems.