Express Integral as a Riemann Sum and solve

**Robb** · June 13th 2009, 06:56 AM

Hello MHF,
I have been able to process this question a little bit, but got a bit stuck when i got to a certain point..

Express the integral as a limit of sums;
$\int_{0}^{\pi}\sin5x \ dx$

So by setting $\Delta x=\frac{\pi}{n}$ and $x_{i}=\frac{\pi i}{n}$
Then, $\lim_{n\to\infty}\sum_{i=1}^{n}\sin(\frac{5\pi i}{n})*\frac{\pi}{n}$

$\lim_{n\to\infty}\frac{\pi}{n}\sum_{i=1}^{n}\sin(\ frac{5\pi i}{n}) $
$\lim_{n\to\infty}\frac{\pi}{n}\sin(\frac{5\pi}{n}\ sum_{i=1}^{n}i) $
$\lim_{n\to\infty}\frac{\pi}{n}\sin(\frac{5\pi}{n}* \frac{n^2+n}{2}) $
$\lim_{n\to\infty}\frac{\pi}{n}\sin(\frac{5\pi}{2}* (1+n)) $

But form here I dont know if i have done something incorrectly, or how to keep reducing it to get rid of n..

**Plato** · June 13th 2009, 07:49 AM

Originally Posted by Robb

Express the integral as a limit of sums;
$\int_{0}^{\pi}\sin5x \ dx$
So by setting $\Delta x=\frac{\pi}{n}$ and $x_{i}=\frac{\pi i}{n}$
Then, $\lim_{n\to\infty}\sum_{i=1}^{n}\sin(\frac{5\pi i}{n})*\frac{\pi}{n}$

$\lim_{n\to\infty}\frac{\pi}{n}\sum_{i=1}^{n}\sin(\ frac{5\pi i}{n}) $
$\color{red}\lim_{n\to\infty}\frac{\pi}{n}\sin(\fra c{5\pi}{n}\sum_{i=1}^{n}i) $

The part is red above is incorrect. The sine function is not additive.

**Robb** · June 13th 2009, 08:04 AM

Ah yes, that makes sense. Thanks allot for that.
I actually just re-read the question, and it says to use a computer to calculate the sum and the limit... Is there away to analyse trigonometric functions using riemann sums without a computer for exam preperation(ie. can this question be taken any further)?

Thanks.

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