In this video I go over another example on approximate integration and this time use the error bound formulas for the Trapezoidal and Midpoint Rule approximation methods to select the number of intervals needed in the approximations for the integral of the function 1/x from x = 1 to x = 2 to guarantee that the accuracy or error is within 0.0001. This a very useful video in understanding how to use the error bound estimate to make accuracy predictions so make sure to watch this video! And like often is the case the Midpoint Rule needs less intervals in the approximation than the Trapezoidal Rule to maintain the same level of accuracy.
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Approximate Integration: Example 2
Example: How large should we take n in order to guarantee that the Trapezoidal and Midpoint Rule approximations are accurate to within 0.0001 for the following integral?
Solution:
Accuracy within 0.0001 means that the size of the error should be less than 0.0001.
Thus, for the Trapezoidal Rule, we choose n such that:
Important Note: From my earlier videos on the error bounds and their proofs, it is quite possible to choose a lower value of n and still be within the given accuracy but choosing n to be 41 is the smallest value to guarantee that the accuracy is within 0.0001.
For the Midpoint Rule, we choose n such that:
This further shows how the Midpoint Rule is, most often, more accurate than the Trapezoidal Rule. This also shows how, for very large values of n, we can save computing power by using the Midpoint Rule.