Consider the region between the graph of f(x) = 1
x
and the x-axis on the interval [1, 5]. Sketch the region and write the correct form of Riemann Sum using right endpoints and n intervals.

Change in x=

xi=

A=

And two expressions to represent the value of the area. Forms are limit of Riemann Sum and definite integral.

This is a good example to look at the consequences of a Riemann sum. The right endpoint formula will intinsically give you an overestimate because the right point touches the actual function. The summation will look like this:

Sum((1+dx*n)*dx) with n starting at 1 and ending at a number decided by the x-step size. For example, with a dx value of 0.2, the stop integer of the sum is given by (5-1)/0.2 which turns out to be 20.

As you can see, the sum is representative of adding many discrete rectangles. In this example, the sum can be evaluated by plugging in the values.

Sum((1+0.2*n)*0.2) n=1:20 which gives 12.4

It can be seen by integrating over this interval that the exact sum is 12. It can also be observed that the limit of the summation as dx approaches zero is 12, which is a good check for what we would expect. Hope this helps!

Apr 28th, 2015

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