Question

need help with simpsons rule, paul's online notes and wolfram seem to be producing two different answer

Answer 1

so the function I am looking at is here, simpsons rule
http://tutorial.math.lamar.edu/Classes/CalcII/ApproximatingDefIntegrals.aspx

Answer 2

using n = 4, approximate integral exp(x^2), x =0..2 using simpsons rule.
paul got 17.3536
wolfram got a different answer

Answer 3

i dont see any problem with paul's work

Answer 4

wolfram got 16.5
http://www.wolframalpha.com/widget/widgetPopup.jsp?p=v&id=174a81e7a9ffb5aed0a790093981aaab&title=Simpson%27s%20Rule%20Calculator%20MyAlevelMathsTutor&theme=blue&i0=e^%28x^2%29&i1=0&i2=2&i3=.5&podSelect=&podstate=SymbolicForm__Show%20details

Answer 5

you might have to put that function into wolfram's widget

Answer 6

http://www.wolframalpha.com/input/?i=integrate++e^%28x^2%29+using+simpsons+rule+from+0+to+2+using+2+intervals

Answer 7

The total number of intervals used in Simpsons rule, is half the number you're actually given, because of how it works (in each interval you're giving 3 points, instead of 2).

Answer 8

(Because you're approximating with quadratics, and not with lines)

Answer 9

And you can actually see that from the diagram shown. You have 7 points (X0, X1, X2, ... , X7) that correspond to 6 intervals. But you're only actually using 3 (The green, yellow and blue ones).

Answer 10

here http://www.wolframalpha.com/input/?i=integrate++e%5E%28x%5E2%29+using+simpsons+rule+from+0+to+2+using+4+intervals
i fixed it

Answer 11

now plug the same function into this calculator
http://nastyaccident.com/calculators/calculus/simpsonsRule

Answer 12

also if you scroll down to pauls, he gets 17

Answer 13

look, it should be 1/6 in front, not 1/12

Answer 14

Yeah, I know. I'm just saying that Wolfram defines intervals differently. If you use half the number of intervals for wolfram, you get exactly the same answer.

Answer 15

Wolfram counts the amount of colors in
http://tutorial.math.lamar.edu/Classes/CalcII/ApproximatingDefIntegrals_files/image003.gif
Which are 3.
While your other calculator, counts the amount of intervals (Which are twice the amount of colors).

Answer 16

sorry i dont understand

Answer 17

hmm

Answer 18

the subinterval should be
[0,.5] [.5,1] [1,1.5] and [1.5,2]

Answer 19

When you use two intervals in wolfram, h = 0.5. Which Is the same distance between subintervals you're using in the other calculator.

Answer 20

T = .5/3 ( f(0) + 2f(.5) + 4f(1) + 2(f1.5) + f(2))

Answer 21

well, i get the same answer if i use n = 8 in this calculator
http://nastyaccident.com/calculators/calculus/simpsonsRule

Answer 22

When you use n=4 in wolfram?

Answer 23

perl do you know differential equations? or calculus iv?

Answer 24

hi usuki , yes i know it

Answer 25

lessis, ok so for some reason n=2 in wolfram is n=4 for paul

Answer 26

In your calculator, the n you use must be twice the n you use in wolfram. Because of how they are definind their intervals.

Answer 27

thank gawd need helpuuu

Answer 28

we're pulling all nighters :D

Answer 29

lessis, this is how wolfram defines it http://mathworld.wolfram.com/SimpsonsRule.html

Answer 30

Wolfram Intervals = n
Pauls Intervals = 2n

Answer 31

yeah, why is that?

Answer 32

i would say its the other way around, wolf = 2n , paul = n

Answer 33

wolf's is more accurate

Answer 34

I'm cold brrr

Answer 35

They're defining intervals differently. Paul is using half intervals, for some weird reason.

Answer 36

2 intervals in wolfram is the same as 4 with Paul
4 intervals in wolfram is the same as 8 with Paul
n intervals in wolfram is the same as 2n with Paul

Answer 37

hmm, oh i see something on line (9)
An extended version of the rule can be written for f(x) tabulated at x_0, x_1, ..., x_(2n) as

Answer 38

vector field?

Answer 39

so wolfram divides [a,b] into 2n , and paul divides it into n?

Answer 40

its not so weird, if i found another calculator that does that

Answer 41

so you were saying, the reason for using 2n is to ensure that there are an even number of points (you dont want to be stuck with an odd number of points)

Answer 42

Yes. That's why the total number of intervals Wolfram uses (2n) is the same as the number of intervals Paul uses (which paul just calls n).

Answer 43

is there such a condition on the trapezoid rule?

Answer 44

Yeah, that's Pauls idea basically. With wolfram you can use uneven number of intervals, because it's using twice the number you give it.

With the trapezoid rule, you can use any integer number of intervals you want, because you're not forming quadratics.

Answer 45

and quadratics need at least 3 points (or 2 subintervals)

Answer 46

, approximating by quadratics need 2n subintervals, (2n+1 points)

Answer 47

Yes!

Answer 48

well you solved the mystery :)

Answer 49

haha, i feel stupid

Answer 50

well maybe i will notify paul, ive written to him in the past

Answer 51

It's kinda weird, honestly. Especially since Paul isn't really using the intervals. (Just the sub-intervals)

Answer 52

I like the way Wolfram asks for input better, because It's basically asking you for the number of quadratics you want to use to approximate the curve.

Answer 53

yeah im a bit confused now, so a subinterval is actually one quadratic subinterval

Answer 54

guys calc iv....plz

Answer 55

oh i get it now, like the trapezoid is a trapezoid. a quadratic a quadratic

Answer 56

so a subinterval in simpsons goes through three points, ok . i see ., thanks again

Answer 57

calc iv? perl?

Answer 58

but the calculator is wrong then ?
plug in the function http://nastyaccident.com/calculators/calculus/simpsonsRule

Answer 59

ok usuki,

Answer 60

XD nasty accident.com not a good name for a site

Answer 61

It's not wrong. It's just asking for a different thing. Wolfram asks for the number of intervals, which can be any integer (1, 2, 3, 4, etc), while Paul asks for the number of subintervals, which has to be a pair.
(Which, when doing everything from scratch, I guess makes more sense, considering how you have to first give the subintervals, to form the quadratics in each interval, to calculate the area)

Answer 62

ahem perlll....