Chapter 6: Techniques of Integration
Section 6.7: Numeric Methods
If λ is the "actual value" of the definite integral in Example 6.7.1, determine empirically (trial-and-error) the smallest partition for which λ and the value given by Simpson's rule agree when rounded to four places.
From Example 6.7.1, λ=5.078061188, and rounded to four decimal places, λ becomes λ^=5.0781. By trial-and-error, the smallest value of n in Simpson's rule for which the approximation to the integral rounds to λ^ is n=10. Simpson's rule with this partition returns 5.078118675, which rounds down to λ^=5.0781.
tutor could be used to implement Simpson's rule for different values of the partition n. Alternatively, the ApproximateInt command can be use, as in Table 6.7.5(a).
Tools≻Load Package: Student Calculus 1
Context Panel: Assign to a Name≻F
1+sinxlnx+1→assign to a nameF
Apply the ApproximateInt command
ApproximateIntF,x=1..4.0,partition=8,method=simpson,partitiontype=normal = 5.078201325
ApproximateIntF,x=1..4.0,partition=10,method=simpson,partitiontype=normal = 5.078118675
Table 6.7.5(a) Determining n for which Simpson's rule agrees with λ when rounded to four places
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