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multiple integal

Posted on 2008-05-01 18:09 By casperyc (Maple Rating 2 133)
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hi

i am doing a multiple integal integration from one of my tutorial sheet

 

Given D is the triangular region whose vertices are (0, 0), (Pi,0), (Pi,Pi), find

Int(Int(x*cos(x+y),x),y);

 

then I worked out something like

 

Int(Int(x*cos(x+y),y=0..x),x=0..Pi);value(%);                                         

 

#eqn#

 

the answer gives me a NEGATIVE value..

 

if i do it but hand i get something at last

Int(-x*sin(2*x)+x*sin(x),x=0..Pi);value(%);

 

-3/2*Pi

 

which is as expected

 

 

what;s wrong with my equation ?? #eqn#

‹ Sounds simple, but I get errors for trying to control DSOLVE,NUMERIC stepsize?!?!?!?! Fixed pt iteration ›

ooops

Comment on 2008-05-01 18:11 from casperyc (Maple Rating 2 133)

doing by hand the last step gives me

 

Int(-x*sin(2*x)+x*sin(x),x=0..Pi);value(%);

 

gives me POSITIVE value which is as expected

 

what's wrong with

 

Int(Int(x*cos(x+y),y=0..x),x=0..Pi);value(%);

Robert Israel's picture

Negative

Comment on 2008-05-01 18:54 from Robert Israel (Maple Rating 4 1248)

Maple's answer of -3*Pi/2 is correct.

int(x*cos(x+y), y = 0 .. x) = -x*sin(x)+x*sin(2*x)

(an antiderivative is x*sin(x+y))

int(-x*sin(x)+x*sin(2*x), x=0..Pi) = -3*Pi/2

(an antiderivative is -sin(x)+x*cos(x)+1/4*sin(2*x)-1/2*x*cos(2*x))

 

yes i know

Comment on 2008-05-01 19:40 from casperyc (Maple Rating 2 133)

so for this question

i wrote down the wrong equation? wrong limits?

more about integrals

Comment on 2008-05-01 19:47 from casperyc (Maple Rating 2 133)

Int(exp(1)^(-x^2),x=0..infinity);value(%);

 

 

maple returns me with an answer of

 (1/2)*sqrt(Pi)

 

however my tutorial sheet says it should be sqrt(Pi)

 

which is correct?

 

thanks

lemelinm's picture

The answer is

Comment on 2008-05-01 20:58 from lemelinm (Maple Rating 2 164)

In the "Table of integrals, series and products" of Gradshteyn and Ryzhik, page 307 the answer is

Int(exp(-x^2),x=0..infinity)=sqrt(Pi)/2

and since Maple give the same answer, my response is that your tutorial sheet is wrong.

mario.lemelin@cgocable.ca

okie this is cool

Comment on 2008-05-02 05:25 from casperyc (Maple Rating 2 133)

how about my orginial question?

why do i get i negative value?

did i put down the wrong limits of intergration?

 

thanks

 

chen

Robert Israel's picture

Why not?

Comment on 2008-05-02 13:03 from Robert Israel (Maple Rating 4 1248)

Why shouldn't you get a negative value? You should get a positive value when you integrate a positive function over some region. But here the function is positive in some places and negative in others, so there's no reason to expect the result to be positive.

i see

Comment on 2008-05-02 16:35 from casperyc (Maple Rating 2 133)

it's started this topic for 2 days ,so.... not really get into too much for it

Table of integrals, series and products

Comment on 2008-05-02 05:52 from casperyc (Maple Rating 2 133)

and just curious

 

"Table of integrals, series and products" of Gradshteyn and Ryzhik

 

where can i get this ? is there a free PDF copy ?

 

thanks

link

Comment on 2008-05-02 11:03 from Joe Riel (Maple Rating 4 1118)

www.mathtable.com/gr/

it's only a 7-page errata

Comment on 2008-05-02 11:45 from casperyc (Maple Rating 2 133)

how do i get to the actual document?

is it not free?

 

thanks

Axel Vogt's picture

the usual way is ...

Comment on 2008-05-02 17:05 from Axel Vogt (Maple Rating 4 887)

The usual way is ... to buy it ... or get it from a library.

You may accept that not every thing is for free ... of course only on a legal basis ...

lemelinm's picture

Unfortunately

Comment on 2008-05-02 12:22 from lemelinm (Maple Rating 2 164)

it is not free but it is a wonderfull book to have.  It's a little bit pricey thou.  The link of Joe give it all.

 

About your firts question, I have

%=value(%)

In a mathematical point of view, the answer is correct.  But in a practical point of view, I say to my students to take the absolute value since there is no sense in a negative value

mario.lemelin@cgocable.ca

ok thanks

Comment on 2008-05-02 12:28 from casperyc (Maple Rating 2 133)

yep i just had my tutorial and it is suggested that to take the abs value.

 

about the book, it's too pricey for me at the moment. i will be how things will go on in the second year

 

thanks to all of you

 

chen

Free

Comment on 2008-05-03 04:46 from Mariner (Maple Rating 4 562)

Chen,

You might find http://integrals.wolfram.com/index.jsp useful. It's free and you will need pencil, paper and a little thought to get the answers you want.

For example, 

Int(exp(x^(-2), x) is given as (1/2)*sqrt(Pi)*erf(x).

Since erf(infinity) = 1 and erf(0)=0, the answer you want is

(1/2)*sqrt(Pi)

Hope you find this helpful.

J. Tarr

however..polar coordinates

Comment on 2008-05-03 09:53 from casperyc (Maple Rating 2 133)

however, our tutoial sheet instructed us to use ploar coordinates to calcultate it

 

on the onther hand, i will have a look at it,

 

thank you!

 

chen

in Wikipedia

Comment on 2008-05-04 15:43 from jakubi (Maple Rating 4 692)

http://en.wikipedia.org/wiki/Gaussian_integral

exactly

Comment on 2008-05-05 11:40 from casperyc (Maple Rating 2 133)

thank for this

lemelinm's picture

MapleNet site

Comment on 2008-05-03 11:13 from lemelinm (Maple Rating 2 164)

I found a link to a maplenet application that is suppose to give the result of a integral that you want.

maplenet.maplesoft.com/maplenet/worksheet/Integrals/Integrals.mw;jsessionid=9C5C4CC42E06FD1EFB047906DA448765.mn001

But it is very disturbing to see that it cannot do the integration of

int(exp(-q^2*x^2),x)

while Mathematica can.  Maple should develop a site in wich a student can make derivative, integration and plotting of functions.  Maybe someone know such a link?

mario.lemelin@cgocable.ca

what's the -q^2?

Comment on 2008-05-03 12:01 from casperyc (Maple Rating 2 133)

hey there,

you said maple can't do integration of

 

Int(exp(1)^(-q^2*x^2),x) ;

? what's the -q^2 there?

have you tried like numerically?

 

what's the -q^2?

Comment on 2008-05-03 12:02 from casperyc (Maple Rating 2 133)

hey there,

you said maple can't do integration of

 

Int(exp(1)^(-q^2*x^2),x) ;

? what's the -q^2 there?

have you tried like numerically?

 

exp

Comment on 2008-05-04 21:31 from Joe Riel (Maple Rating 4 1118)

Use exp(x), not exp(1)^x (though Maple handles either form with this integral).  However, exp(x) is not, in general, the same as exp(1)^x.  

 int(exp(-q^2*x^2),x);
                                    1/2
                                  Pi    erf(q x)
                              1/2 --------------
                                        q

This site is kind of beta

Comment on 2008-05-03 14:39 from jakubi (Maple Rating 4 692)

On clicking on this site, I have received warnings and error messages from http://maplenetbeta.maplesoft.com.

And sometimes I have got it produce an answer, but  sometimes not. In  particular, with your integral, I  have  received  on occasions  this error  message:

The page at http://maplenet.maplesoft.com says:
Errors occured while processing the latest request:
(inPreprocessorTab[_XML_mrow]) invalid subscriptor selector

java

Comment on 2008-05-03 21:45 from casperyc (Maple Rating 2 133)

i think java is what you need

certainly not

Comment on 2008-05-03 21:52 from jakubi (Maple Rating 4 692)

without java it would not work at all.

gkokovidis's picture

Int

Comment on 2008-05-03 12:35 from gkokovidis (Maple Rating 4 618)

>restart:

>Int(exp(-q^2*x^2),x) ;

Int(exp(-q^2*x^2),x)

>value(%);

Pi^(1/2)/q*erf(q*x)

 

Regards,
Georgios Kokovidis
Dräger Medical

are you sure?

Comment on 2008-05-03 12:46 from casperyc (Maple Rating 2 133)

why do i get HALF of your value?

i did it in Classical mode

gkokovidis's picture

Trust but Verify

Comment on 2008-05-03 12:57 from gkokovidis (Maple Rating 4 618)

When I copied the output, I did not highlight the 1/2.  Your answer is correct.

1/2*Pi^(1/2)/q*erf(q*x)

 

Regards,
Georgios Kokovidis
Dräger Medical

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