Chapter 6: Techniques of Integration
Section 6.7: Numeric Methods
Use the error bound in Table 6.7.1 to estimate the value of the partition n for which Simpson's rule makes an absolute error of no more than 10−4 when estimating λ, the value of the definite integral in Example 6.7.1.
What is the actual value of n for which Simpson's rule achieves this accuracy?
If fx=1+sinxlnx+1, write h4180 maxx(|f″|) b−a, the error bound in Table 6.7.1, as b−a5180 n4maxx(|f″|) and solve the inequality 4−15180 n4M≤10−4 for n, where M=maxxf4≐1.0443. This gives n≐± 10.9, so the appropriate value is n=12. However, for n=10, Simpson's rule actually approximates λ with an error no worse than 10−4.
Determine M, the maximum of the absolute value of the fourth derivative of the integrand over the interval of integration.
Context Panel: Assign Function
fx=1+sinxlnx+1→assign as functionf
From Figure 6.7.6(a), a graph of |f4x| on 1,4, estimate:
In fact, f41.0 = 1.044281148
Figure 6.7.6(a) Graph of fx on 1,4
With a=1,b=4,M≐1.0443, solve the inequality b−a5180 n4maxx(|f4|)≤10−4 for n.
Write the inequality b−a5180 n4maxx(|f4|).
Context Panel: Solve≻Solve
The appropriate choice of n is the first positive even integer greater than 10.9. Hence, n=12 guarantees that Simpson's rule will approximate λ with an error of no more than 10−4. From Example 6.7.1, take λ to be the number 5.078061188. To determine the actual value of n for which Simpson's rule approximates λ with the desired accuracy, use the ApproximateInt command as per Table 6.7.6(a).
Tools≻Load Package: Student Calculus 1
Define L as the actual value of the integral.
Use the ApproximateInt command and compare to λ
Table 6.7.6(a) The smallest value of n for which Simpson's rule approximates λ to within 10−4
By experiment, it is determined that n=10 is the smallest value of n for which Simpson's rule approximates λ with an error no worse than 10−4.
Note that for Simpson's rule, the ApproximateInt command defaults to the subinterval form of the partitiontype option. This form of the rule evaluates the integrand at 2 k+1 points, thereby obtaining a much greater accuracy. To keep the number of function evaluations to k+1, it is essential to include the normal form of the partitiontype option.
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