My calc. AB teacher decided against teaching Riemann sums with sigma notation and as I have been studying BC independently I sort of came up with this problem. The answer is pi of course, but if I use the sigma notation and add "infinite rectangles" I need to have some way of coming up with interval widths in a different manner than the normal (b-a)/n. the largest problem seems to be that I don't know how to figure out how to evaluate this type of integral where the width changes. On top of this I know pi is irrational so I figure my answer should come out to an irreducible infinite series but I haven't a clue how to come to that point using these riemman sums. Definite integration is not much problem, I can use arcsin and tables I think, I just want to better understand this method and my teacher hasn't taught anything above math heuristics for a long time. Thanks!
The problem is that practically calculating Riemann Sums even in the more tractable sense taught in calculus is damn near impossible for most functions. This is why one has the Fundamental Theorem of Calculus, which does this in general for continuous functions. One may hope to follow the proof the FTC for a clue how to do this, but unfortunately it uses existence tricks etc. to circumvent any real ugliness. Thus, in general something like is not a function conducive to using Riemann sums. Now, if you wanted to do that'd be a different story
Once again, you have completely misread the question! The question was about using Riemann sums to approximate an integral, not about doing the integral by finding an anti-derivative. While you are usually very helpful, occaisionally, you go off on strange tangents! Please read the problems more carefully.
I changed the integral as an idea for the Riemann Sum. Don't always jump to conclusions unless you ask about my intentions first, please.
Edit:
As you can see (http://mathworld.wolfram.com/RiemannSum.html), it will work.