Applying Simpson's Rule with 7 Pieces: Solving the Int.

[shamieh]

When applying Simpson's rule, suppose I have to slice up my function into 7 pieces, which would be odd. Then how would I apply Simpson's rule to my problem? Doesn't the n have to be equal to some even number?

I'm a bit confused, here is the question.

Estimate \(\displaystyle \int ^{2\pi}_0 f(x) \, dx\)The following data was collected about a function f(x)
\(\displaystyle x|f(x)\)

\(\displaystyle 0 | 1.000\)

\(\displaystyle \frac{\pi}{3} | 1.513\)

\(\displaystyle \frac{2\pi}{3} | 0.696\)

\(\displaystyle \pi | 1.000\)

\(\displaystyle \frac{4\pi}{3} | 1.107\)

\(\displaystyle \frac{5\pi}{3} | 0.937\)

\(\displaystyle 2\pi | 1.000\)

Sorry if it looks sloppy, I don't remember how to draw tables on this forum, anyways, with that being said, They are giving me 7 values, so they want me to split it into 7 pieces correct? So they want \(\displaystyle \frac{\Delta x}{3} [ f(0) + 4(\pi/3) + 2( 2\pi/3) ... 2\pi]\)

By the way I am just guessing they want me to split it into 7 pieces. Maybe they just want 6? I'm just not too sure, I feel like the approximate area would need to include all values. Thanks in advance for your help

 

[MarkFL]

You do have 7 data points, but only 6 intervals ($n=6$), so Simpson's Rule will work here with no problems. :D

 

[shamieh]

Oh I see. I'll be back with my solution soon!

 

[shamieh]

Can someone verify my solution? It looks reasonable, just want to make sure though.(Muscle)

So I ended up with \(\displaystyle [1.000 + 4(1.513) + 2(0.696) + 4(1.000) + 2(1.107) + 4(0.937) + 1.000]\)

Which turned out to be \(\displaystyle \approx 19.406\)

Then I did \(\displaystyle \frac{\pi}{9} \approx .34906585039\)

thus \(\displaystyle .34906585039 * 19.406 \approx 6.77397189284\)

 

[MarkFL]

Looks good to me! :D