Consider the Riemann sum of the function f(x) = cos(2x) on the interval (0,pi/2) with
n = 3 equal size subintervals, that uses the right endpoint of each subinterval as the sample
a) Write down the above sum using sigma notation; do not simplify or evaluate.
b) Evaluate the sum; simplify, but do not approximate.
Please help my teacher spent like 5 minutes on this in class and now its on our assignment.
Thanks in advance.
Nov 13th 2012, 05:08 PM
Re: Reimann sums
i will tel you HOW to do this, but you will have to actually DO it yourself.
we have an interval [0,π/2], and we are going to chop it up into 3 equal pieces.
this will becomes the THREE intervals [0,π/6], [π/6,π/3] and [π/3,π/2].
so our right-hand endpoints are:
π/6, π/3 and π/2.
we are going to approximate the area between the curve f(x) = cos(2x) and the x-axis, by three rectangles:
the rectangles will have HEIGHT f(xi), where xi is one of our 3 endpoints, and each rectangle will have a BASE (width) of length (1/3)(π/2) = π/6.
so we're going to have 3 terms in our sum, each one will look like:
cos(2xi)(π/6). i leave it to you to put this in "sigma" notation.
the GENERAL form of a riemann sum from a to b is:
where x1 = a, and xn+1 = b, and for each i, xi < xi+1.
often, the sub-intervals [xi,xi+1] are all chosen to be the same size, while xi* is allowed to be ANY point in [xi,xi+1].
it is, however, customary to pick a sure-fire way of coming up with some "organized" way of picking the xi* (such as left endpoint, right endpoint, midpoint, maximum value or minimum value). if f is a continuous function, and n is large enough, it won't make "too much difference" which point we pick, because all these values will be "close together".