Approximate Integration: Example 1: 1/x

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In this video I go over an example on approximate integration and approximate the integral of the function 1/x from x = 1 to 2 using 5 subintervals, or n = 5, and using the Trapezoid and Midpoint Rules. This is an interesting example because we can easily check the accuracy of both approximation methods because the exact value of the integral is ln(2). Contrary to initial visual appearance of both methods, the Midpoint Rule is about two times more accurate than the Trapezoid Rule. I will explain the reason why in detail in my later videos so stay tuned!


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Approximate Integration: Example 1

Approximate Integration Example 1.jpeg

Use (a) the Trapezoidal Rule and (b) the Midpoint Rule with n = 5 to approximate the integral:

In this example, we deliberately chose an integral whose value can be computed explicitly so that we can see how accurate the Trapezoidal and Midpoint Rules are:

The error in using an approximation is defined to be the amount that needs to be added to the approximation to make it exact:

From our example, we see that the errors in the Trapezoidal and Midpoint Rule for n = 5 are:

As I explained in my earlier video, for most cases the Midpoint Rule is more accurate than the Trapezoid Rule even though it does not easily appear this way. I will further explain the reason why in my next video so stay tuned!



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