A simple overview of the Riemann sum

By Guillaume Pelletier on January 5, 2009

I was reading ahead through my math book tonight and I saw in the introduction of integrals a section on Riemann sums. Hey, what a great occasion to brush up on my LaTeX skills! Here it goes:

The approximation method for calculating the area under a curve is named after Bernhard Riemann, who was one of the pioneers of modern integration theory.

If we take a function f so that y=f \left( x \right) on an interval \left[ a,b \right], where f is a continuous function and 0\leq f \left( x \right) :

  1. We divide the interval in n subsections of width equaling \Delta x={\frac {b-a}{n}}. This operation is called a partition of the interval \left[ a,b \right].
  2. We evaluate f at the right end-point \left( a+k\Delta x \right) of the kth subsection for k = 1, 2, ..., n.
  3. we write the sum of areas of the n rectangles, which we write:

    S_{{n}}=\sum _{k=1}^{n}f \left( a+k\Delta x \right) \Delta x

  4. Since we expect the approximation of S_{{n}} to get more and more accurate as \Delta x gets smaller, we say that the area A delimited by the curve, the x axis and the lines of equations x = a and x = b are equal to the limit of S_{{n}} when \Delta x\mapsto 0 (or n \mapsto + \infty). We then write

    A=\sum _{k=1}^{\infty }f \left( a+k\Delta x \right) \Delta x
    If that limit exists. this means we can approximate A with an arbitrary degree of precision with \Delta x small enough, or n large enough.

Wow, this is going to be a long term for me, seeing as I already did all this stuff back in highschool. At any rate, at this point I should probably try to focus more on chemistry, or even biology.

Anyway, that’s it for today, I might add images to this article later if I feel like it, or even some variations of the Riemann method of approximation, like the middle-sum or the trapezoidal rule… I don’t think it is absolutely critical information however, so I realize it’s probably not worth the effort.

Posted in University