In this video I go over a very useful example on applying the Simpson's Rule error bound formula to determine the adequate number of sub-intervals needed to ensure a given accuracy, in this case being within 0.0001. The integral being approximated is the same the integral of the function 1/x from x = 1 to x = 2. This is the same function as in example 2 which was approximated using the Trapezoidal and Midpoint Rules. This video shows how the Simpson's Rule can achieve the same accuracy at a far less number of sub-intervals. This means that a calculator or computer program would need less calculations and thus less computing power to achieve a high level of accuracy. In very advanced mathematical and physics applications, such as fluid dynamics, and airplane modelling, lowering the required computing power is of utmost importance. I go over briefly about computing and mathematical approximation in this video and it would be pretty interesting for you watch this video and learn from.
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Approximate Integration: Example 6: Simpson's Rule Error
How large should we take n to guarantee that the Simpson's Rule approximation for:
to accurate to within 0.0001?
Solution:
Compare this with Example 2, where it was shown that to achieve this same accuracy using the Trapezoidal Rule we needed n = 41 and for the Midpoint Rule we needed n = 29.
Thus we can achieve greater accuracy while using less intervals by using the Simpson's Rule, and the less intervals required can greatly reduce the number of calculations and thus the computing power required.
Many calculators and computer algebra systems have a built-in algorithm that computes an approximation of a definite integral. Some of these machines use Simpson's Rule; others use more sophisticated techniques such as adaptive numerical integration. This means that if a function fluctuates much more on a certain part of the interval than it does elsewhere, then that part gets divided into more subintervals. This strategy also reduces the number of calculations required to achieve a prescribed accuracy.