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**preview**shows half of the first page. to view the full**1 pages of the document.**7.7 Integral Techniques

Integral Approximation

Question #4 (Medium): Integral Approximation & Error Bound Using the Midpoint Rule

Strategy

The error bounds are given by:

, where , and for the Midpoint

Rule. Similarly for Trapezoidal Rule:

, where , and . Notice that

only the denominator is slightly different. As for Simpson’s Rule:

, where , and

, meaning the fourth derivative of the function is bound by factor. Usually the question

provides these input values in order to calculate the error bounds.

Sample Question

1) Given the dataset, estimate the value of the integral

using the Midpoint Rule.

2) If , estimate the error involved in the approximation from part 1).

Solution

1) Integral is approximated using the Midpoint Rule:

. Pick the midpoints (ie. every alternating points starting from

the second, ending with the second last), then . Thus:

2) Error bound for the Midpoint Rule is:

, where , and . Then

in this case , since

, and since , , then:

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