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I have to use the Composite Simpson's Rule to approximate the integral $\int_0^1 t^2\cdot sin(\frac{1}{t}) dt$. I've used the Composite Simpson's Rule, but when I work through the algorithm, one step is throwing me off.

When I try to compute $XI0 = f(a) + f(b)$, $f(a)$ is $f(0)$, which is undefined ($0\cdot sin(\frac{1}{0})$). How am I supposed to work through this? All the other steps work fine.


For reference, here is the Algorithm listed in my textbook, Numerical Analysis Ninth Edition, by Richard L. Burden and J. Douglas Faires:

To approximate the integral $I = \int_a^b f(x) dx$

  • INPUT endpoints $a$, $b$; even positive integer $n$
  • OUTPUT approximation $XI$ to $I$
  • Step 1 Set $h = \frac{b-a}{n}$
  • Step 2 Set $XI0 = f(a) + f(b)$; $XI1 = 0$; $XI2 = 0$
  • Step 3 For $i = 1,...,n-1$ do Steps 4 and 5
    • Step 4 Set $X = a + ih$
    • Step 5 If $i$ is even, set $XI2 = XI2 + f(X)$; else set $XI1 = XI1 + f(X)$
  • Step 6 Set $XI = h(XI0 + 2\cdot XI2 + 4\cdot XI1)/3$
  • Step 7 OUTPUT(XI); STOP
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1 Answer 1

up vote 1 down vote accepted

You can figure out what the value 'should' be here to get the algorithm to converge optimally. It's best when the function is continuous, so we ask what the limit of $t^2 \sin \frac{1}{t}$ is as $t\to 0$. Then it smoothly joins on.

I'll leave this as an exercise, but if you want a hint...

Hint: $|\sin u| \le 1$ for all $u$.

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So it's $0$, right? I couldn't figure it out (before I got this answer), and just left that part as $0$, and the answer matched the back of the book. It's because we don't need to worry about $sin{\infty}$, because it will always be between $-1$ and $1$, and so when multiplied by $0$, it will be $0$? –  SSumner Apr 22 '13 at 1:14
1  
Yes, essentially! More mathematically, avoiding saying $\sin\infty$, we say that given any $\epsilon>0$, there exists $\delta>0$ such that $|f(x)| < \epsilon$ for all $|x|<\delta$. This is the definition of the limit being 0. –  Sharkos Apr 22 '13 at 9:03
    
Excellent, thanks! –  SSumner Apr 22 '13 at 16:58

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