MAT136H1 Lecture Notes - Trapezoidal Rule
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7.7 Integral Techniques
Question #4 (Medium): Integral Approximation & Error Bound Using the Midpoint Rule
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.
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).
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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