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 Trapezoid rule's value 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 the Trapezoid rule for which the approximation to the integral rounds to λ^ is n=312. The Trapezoid rule with this partition returns 5.078050007, which rounds up to λ^=5.0781.
tutor could be used to implement the Trapezoid rule for different values of the partition n. Alternatively, the ApproximateInt command can be use, as in Table 6.7.2(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=311,method=trapezoid,output=value = 5.078049935
ApproximateIntF,x=1..4.0,partition=312,method=trapezoid,output=value = 5.078050007
Table 6.7.2(a) Determining n for which the Trapezoid rule agrees with λ when rounded to four places
At n=311, the Trapezoid rule yields 5.078049935, which rounds to 5.0780. On the other hand, at n=312, the Trapezoid rule yields 5.078050007, which rounds to 5.0781=λ^.
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