computing riemann's sums using Fundamental Thm: lim[n->infty]1/n Σ sin (k pi/2n)

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computing riemann's sums using Fundamental Thm: lim[n->infty]1/n Σ sin (k pi/2n)

the question is:
Identify the following limits as limits of Riemann Sums, and use definite integrals (and the Fundamental Theorem of Calculus) to compute them.


I assume this means that they want me to convert the limit into a definite integral,

(a) lim 1/n Σ sin (kpi/2n)
n→∞

b n
i know that ∫ f(x) dx = lim Σ f(a+k(b-a/n)(b-a/n)
a n→∞ k=1

so by just looking at my function i'm guessing that a=0
and b-a/n = pi/2n
so that b=pi/2

and f(x) probaby= sinx

pi/2
∫ sinx dx
0
but this does not look right, and I am unsure what I am missing/ how to approach the problem.

(i apologize for the terrible formatting...)

 

[mmm4444bot]

I apologize for the terrible formatting

We cannot "draw" formatting at this site because the forum strips out repeated spaces. The work around is to use the code tags to prevent this, as well as a fixed-width font for lining everything up.

It's easier to use established keyboard conventions, for representing things like summations and integrals.

:idea: In the forum guidelines, there is a link for 'Formatting Math as Text'. Check it out. :cool:

 

[Dr.Peterson]

I think you mean this:

(a) \(\displaystyle \displaystyle \lim_{n\rightarrow \infty}\dfrac{1}{n}\sum sin\left(\dfrac{k\pi}{2n}\right)\)

i know that \(\displaystyle \displaystyle \int_{a}^{b}f(x)dx=\lim_{n\rightarrow \infty}\sum_{k=1}^{n}f\left(a+k\left(\dfrac{b-a}{n}\right)\right)\left(\dfrac{b-a}{n}\right)\)


so by just looking at my function i'm guessing that a=0
and (b-a)/n = pi/(2n)
so that b=pi/2

and f(x) probably= sinx

\(\displaystyle \displaystyle \int_{0}^{\pi/2}\sin x dx\)

but this does not look right, and I am unsure what I am missing/ how to approach the problem.

But what are the bounds of the given summation?

It is appropriate to start by just guessing what the function might be, as you did; the next step is to go beyond feeling wrong and actually write out what the summation would be for the integral you wrote, and see whether it is what was given. If not, think about what you might have to change (probably in the argument to the sine). Let me know what you get when you do this.

By the way, if you don't want to use LaTeX formatting as I just did (which took some work!), you can use ideas found here to format almost as readably using plain text.