Expert: Paul Klarreich - 2/18/2007

Question
Integration by Simpson's Rule :  People who sail ships at sea used dead reckoning to calculate distance a ship has gone. Suppose a ship is maneuvering by changing speed rapidly. The table shows it speeds at 2 min intervals. (A knot is a nautical mile per hour. A nautical mile is about 2000 yd) Use Simpson's Rule to find the distance traveled in the 12 min time interval.
Time(min)            Speed(kn)
  0                    33
  2                    25
  4                    27
  6                    13
  8                    21
  10                   5
  12                   9

So, I used the formula for Simpson's Rule and got a number of 310. I know that I need to multiply this by 2/3. I'm not sure how to use the 2000 to get the distance with what I have already done. That's where I am at.  Thanks.

Answer
Questioner:   Colleen
Category:  Calculus
 
Question:  Integration by Simpson's Rule :  People who sail ships at sea used dead reckoning to calculate distance a ship has gone.

>> Just as well -- all those people are dead now, anyway.

Suppose a ship is maneuvering by changing speed rapidly. The table shows its speed at 2 min intervals. (A knot is a nautical mile per hour. A nautical mile is about 2000 yd) Use Simpson's Rule to find the distance traveled in the 12 min time interval.
Time(min)            Speed(kn)
 0                    33
 2                    25
 4                    27
 6                    13
 8                    21
 10                   5
 12                   9

So, I used the formula for Simpson's Rule and got a number of 310. I know that I need to multiply this by 2/3. I'm not sure how to use the 2000 to get the distance with what I have already done. That's where I am at.  Thanks.
..................................
Hi, Colleen,

I don't think you have to multiply by 2/3, I think it's by 1/3.  And I don't think you necessarily use the 2000, but I DO think you have to use a 60.

S.R. says that
{x0+2h
|      f(x) dx ~~ [is approx]
}x0

[Note that the basic interval for S.R. is 2h, not h.]

1/3 h [f(x0) + 4 f(x0+h) + f(x0+2h]

In your example, you have three sections in which to apply the rule, and for each one, h = 2 minutes, which is 1/30 of an hour.  That's what you need so your units (knots = N.M. per hour.) are consistent.

So the calculation goes:(here's your table, again)
Time(min)            Speed(kn)
 0                    33
 2                    25
 4                    27
 6                    13
 8                    21
 10                   5
 12                   9

Section I:     1/3[1/30][33 + 4*25 + 27]
Section II:    1/3[1/30][27 + 4*13 + 21]
Section III:   1/3[1/30][21 + 4*5  +  9]

= 1/90[33 + 100 + 54 + 52 + 42 + 20 + 9]
= 1/90[345]
= 1/30[115]
= 1/6[23], or a little less than 4 nautical miles.
If you like, you can change THAT to yards by multiplying by 2000 (approx).

Did you mess up the arithmetic a bit? (Maybe I did.)  You might note that some numbers get used once (the very first and very last), some get used four times (the ones in the middle of each interval) and some twice (the ones that are shared by two intervals).