Clearly, when we use more subintervals, our rectangles or trapezoids give better and
better approximations to the area under the graph of the function. In the limit as
n gets bigger and bigger, the approximations converge to the exact area under
the curve.
In this Part, we will see that doubling n gives markedly better improvement in
accuracy for some of our approximation methods than it does for others.
Throughout this Part, we will be
approximating the integral of f(x) = 1/x over the interval [1,3]. You will need the
tables of results for this integral you recorded in Parts 1 through 3.
The table below records the errors in the left-hand sum approximation of the integral
of f(x)= 1/x over the interval [1,3] for various numbers of approximating rectangles.
The error is just the approximate value from the table you calculated in Part 1 minus
the exact value of the integral, which is ln(3) = 1.098612289.
The third column gives the improvement ratio in the approximation. For example, the
improvement for n = 10 over n = 5 is error(5)/error(10) = 0.144989/0.069617 = 2.08.
