# Simpsons Rule anyone?

• Jun 8th 2012, 03:50 PM
srh
Simpsons Rule anyone?
The distance, s, moved by a cam follower after 5 seconds is given by

Attachment 24046

Determine an estimate for s using Simpsons Rule with 10 intervals.

I'm not sure about this one at all if anyone can help?

Thanks.
• Jun 8th 2012, 04:39 PM
Reckoner
Re: Simpsons Rule anyone?
Quote:

Originally Posted by srh
I'm not sure about this one at all if anyone can help?

Care to share what you've tried? For reference, Simpson's Rule (and in particular, Composite Simpson's Rule).

Set up your subintervals and then just apply the formula.
• Jun 8th 2012, 04:42 PM
Reckoner
Re: Simpsons Rule anyone?
And, by the way, note that we can evaluate this integral without using numerical methods. Let $\displaystyle u = 7+8t^2\Rightarrow du=16t\,dt$.
• Jun 8th 2012, 08:15 PM
srh
Re: Simpsons Rule anyone?
Sorry, I'm eager to learn but I'm afraid I'm a complete beginner.

I've managed to teach myself some basic calculus but I really need walking through it step by step. :(
• Jun 9th 2012, 03:25 AM
Reckoner
Re: Simpsons Rule anyone?
Quote:

Originally Posted by srh
I've managed to teach myself some basic calculus but I really need walking through it step by step. :(

People are usually more willing to help if you demonstrate that you've made an effort to solve the problem. An easy way to do this is to show any work you've done. And if you can't figure out where to start, you should give specifics on what you don't understand about the process.

I'll set it up this time, though. We want to split [0, 5] into 10 subintervals. That means each subinterval would have to be $\displaystyle \frac{5-0}{10}=\frac12$ units wide. Therefore, the values separating the subintervals are

$\displaystyle \begin{array}{ccc}t_0&=&0 \\ t_1&=&\frac12 \\ t_2&=&1 \\ t_3&=&\frac32 \\ \vdots&&\vdots \\ t_n&=&\frac n2\end{array}$

Now we put these values into the formula.

$\displaystyle \int_0^5t\sqrt{7+8t^2}\,dt$

$\displaystyle =\int_a^b f(t)\,dt$

$\displaystyle \approx\frac{b-a}{3n}\left[f(t_0)+4f(t_1)+2f(t_2)+4f(t_3)+\dots+2f(t_8)+4f(t_ 9)+f(t_{10})\right]$

$\displaystyle =\frac16\left[f(0)+4f\left(\frac12\right)+2f(1)+4f\left(\frac32 \right)+\dots+2f(4)+4f\left(\frac92\right)+f(5) \right]$

Now evaluate the integrand (which I'm calling $\displaystyle f(t)$) at each endpoint.
• Jun 9th 2012, 01:04 PM
srh
Re: Simpsons Rule anyone?
Thanks, I'll give it a go.
• Jun 10th 2012, 05:24 AM
simamura
Re: Simpsons Rule anyone?
First of all here is a link to the Simpson's Rule with shown examples.
Secondly this integral can be solved directly using substitution rule.
Let $\displaystyle u=7+8t^2$ then du=16tdt and $\displaystyle tdt=\frac{du}{16}$.
Since x is changing from 0 to 5 then u is changing from 7 to 207 and your integral now is $\displaystyle \frac{1}{16}\int_7^{207}\sqrt{u}du=\frac{1}{16} \frac23u^{\frac32}|_7^{207}=\frac{1}{24}({207}^{ \frac32}-7^{\frac32})$

Now, if you are still don't understand Simpson's rule than divide interval [0,5] on n=10 subintervals with length $\displaystyle Dx=\frac{5-0}{10}=0.5$
So, $\displaystyle t_i=0+Dx*i$: $\displaystyle t_0=0, t_1=0.5,t_2=1, t_3=1.5,...,t_{10}=1$
Your function is $\displaystyle f(x)=t\sqrt{7+8t^2}$
Therefore, by Simpson's rule approximation is $\displaystyle \frac{Dx}{3}(f(0)+4f(0.5)+2f(1)+4f(1.5)+2f(2)+4f(2 .5)+2f(3)+4f(3.5)+2f(4)+4f(4.5)+f(5))$
Plug values and calculate. For example $\displaystyle f(0)=0*\sqrt{7+8*0^2}=0$