Chapter 6: Techniques of Integration
Section 6.7: Numeric Methods
Table 6.7.1 lists five common fixed-step methods for the numeric evaluation of the definite integral ∫abfx dx. If h=b−a/n, the values of the integrand fx at the evenly-spaced nodes xk=a+k h,k=0,1,…,n, are denoted by fk rather than by fxk. Of course, x0=a and xn=b. The third column provides an upper bound on the absolute value of the error made when evaluating an integral by one of the listed methods. The maximum of the absolute value of the appropriate derivative is taken over x∈a,b.
h224 maxx(|f″|) b−a
h212 maxx(|f″|) b−a
h3f0+fn+4∑k=1n/2f2 k−1+2∑k=1n/2−1f2 k
Table 6.7.1 Common fixed-step methods for numeric evaluation of definite integrals
The Left Endpoint, Right Endpoint, and Midpoint rules are examples of "Rectangular" rules because rectangles are used to approximate the area under the graph of the function being integrated. The Trapezoid rule uses trapezoids, and Simpson's rule approximates the function with a sequence of quadratics and finds the areas under these arcs.
The Rectangular rules evaluate the function at n points, but Simpson's rule (as stated in Table 6.7.1) and the Trapezoid rule evaluate the function at n+1 points.
The traditional statement of Simpson's rule given in Table 6.7.1 requires n to be an even integer. This form of the rule is more easily remembered if described as "h/3 times the sum of the first and last function values plus four times the sum of the odd-indexed values plus two times the sum of the even-indexed values." Maple has both this form and an alternate form in which the function is evaluated at the midpoints of the n subintervals generated by the n nodes. In this form, the function is evaluated at 2 n+1 points, so when making comparisons of the accuracy of the various methods, it is important to do so with methods that expend comparable computational energy.
Use the Trapezoid rule with n=10 to approximate the definite integral ∫141+sinxlnx+1 ⅆx.
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.
Use the error bound in Table 6.7.1 to estimate the value of the partition n for which the Trapezoid 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 the Trapezoid rule achieves this accuracy?
Use Simpson's rule with n=10 to approximate the definite integral ∫141+sinxlnx+1 ⅆx.
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.
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?
Use the Trapezoid rule to approximate the area under the curve determined by the following data points.
Sketch a derivation of the Trapezoid rule.
Sketch a derivation of Simpson's rule.
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