Items tagged with integration

Hi,
   I just finished a math quiz. I needed to find the length of the curve and area of the function, r=3*cos(theta)-2*sin(theta) bounded between 0<=Pi<=2*Pi.
   On the quiz I used Area=int(1/2*(r^2)) dtheta. For the length of the curve I used L=int(sqrt(r^2 + r'(theta)) dtheta.

How do you plug this into Maple and get an answer?
I came up with 20.42.... sq units for the area and 22.65.... for the length of the curve.

Thank you,
Jay.

I have 

dZ(x)=−xdlog(z(x))

where d is the exterior derivative. I would like to recover the function Z(x) by integrating both sides of the equation. How would I compute this in Maple?

int(a(t)*b(t)+2*(diff(a(t), t))*(diff(b(t), t)), a(t));
Error, (in int) integration range or variable must be specified in the second argument, got a(t)
 

do not understand this error message,

how to integrate it?

 Dear All ! 

I really need to solve this problem as soon as possible, As you know the downside equation is not exact, but I can not find its integration factor, blease help me !

                                                                 ∫{ ( ωx + σy ) d x + (ωy −σx)dy}=0

 Regards ,

 

 

so I'm trying this:

restart;

sigma := 0.143e-18;

l_0 := 1.87;

l0 := 1.87;

roll := rand(0 .. 25.0);

f_gauss := proc (x) options operator, arrow; exp(-(1/2)*x^2/`&sigma;_x`^2)/sqrt(2*Pi*`&sigma;_x`^2) end proc;

f_norm := proc (dx) options operator, arrow; int(f_gauss(x), x = -(1/2)*dx .. (1/2)*dx) end proc;

sol_gauss := proc (mix) options operator, arrow; evalf(eval(-ln((int(f_gauss(x)*exp(-2*sigma*N2O*sqrt((1/4)*l_0^2-x^2)), x = -(1/2)*dx .. (1/2)*dx))/f_norm(dx))/(sigma*N2O), [N2O = 0.25e20*mix/100])) end proc;

for ii to 10 do

a := roll();

eval(sol_gauss(a), [dx = l_0, `&sigma;_x` = l0])

end do

After several attempts on this question,

Int(x*sqrt(2*x^4+3),x) with substitution u = sqrt(2)*x^2,

I don't seem to find the solution. Can you guys help me?

So I have an integral that computes perfectly in wolfram alpha but not in maple...

I will post it here

int(1/((4.532055545*10^9/f^4.14-2.311250000*10^5/f^2+(111*(1-0.2163331531e-4*f^2+2.340001656*10^(-10)*f^4))/(1+0.1081665766e-4*f^2)))*(6*10^(-21)*abs(1/f^(4/3)))^2, [f = 50 .. 1500])

the answer should be 3.05364*10^-46

If you try that exact line of code in maple, it will not compute (is stuck on evaluating)


Best Regards to all,
Zeus

Dear all,

I would like to compute numerically using Maple the following improper integral

``

Integrand := (1/4)*(((((6*I)*beta-3-6*C+(6*I)*C*beta)*s^4+((24*I)*C*beta-24*C-12)*s^2+(24*I)*(1+C)*beta)*BesselK(0, s)+12*BesselK(1, s)*(C+1/2)*s^3)*BesselI(1, s)^3+6*BesselI(0, s)*(-(2*(I*beta*C*s^2+(2*I)*beta*C+(2*I)*beta+4*C+2))*s*BesselK(0, s)+((I*beta*C+I*beta-C-1/2)*s^4+((4*I)*C*beta+4*C+2)*s^2+(4*I)*(1+C)*beta)*BesselK(1, s))*BesselI(1, s)^2-(12*(-(1/2*((C+1/2)*s^2+I*beta*C+I*beta+8*C+4))*s*BesselK(0, s)+((I*beta*C+2*C+1)*s^2+(2*I)*(1+C)*beta)*BesselK(1, s)))*s*BesselI(0, s)^2*BesselI(1, s)+6*s^2*((-2*C-1)*s*BesselK(0, s)+BesselK(1, s)*((C+1/2)*s^2+I*(1+C)*beta))*BesselI(0, s)^3)/((BesselI(0, s)^2*s-BesselI(1, s)^2*s-2*BesselI(1, s)*BesselI(0, s))^2*(C+1/2)*s*Pi):

``


However, Maple does seem to give a result for this integral. I have tried to compute from e.g. 0.001 as an approximation but it turns out that the integrand diverges as s goes to zero. I have also tried some options such as method = _d01amc but I get Error, (in evalf/int) powering may produce overflow.

 

I would appreciate it if someone here could provide with some help with regards to the computation of such improper integrals. Thank you.

 

Download question.mw

Hi everybody

In the following attached file, I try to evaluate an integration, K_fL. Unfortunately, Maple does not evaluate it and just puts integration symbol and its bounds. I want to have final integration value. Is there any solution to this integration or maybe Maple can not solve this integration because of the complexity of integrand?

Thanks in advance

Q1.mw

Hi
I want to solve this integration simbolic:


I use this cammand :

But Maple return this:

Would you Please Help me , thanks

I'm running into a very simple problem with the way that Maple integrates Heaviside functions. Naively, it should act like a step function, but it is not integrating properly. See the attached document.

int(int(Heaviside(-x^2-y^2+1), x = -1 .. 1), y = -1 .. 1)

0

(1)

evalf(Int(Heaviside(-x^2-y^2+1), [x = -1 .. 1, y = -1 .. 1]))

3.141592654

(2)

int(piecewise(-x^2-y^2+1 > 0, 1, 0), [x = -1 .. 1, y = -1 .. 1])

Pi

(3)

``


Note that the symbolic integration of the Heaviside function (defined to be 1 inside the unit circle and 0 outside) gives zero, whereas it should clearly give the area of the unit circle, which the numerical integration does. I even checked that the (suposedly equivalent) piecewise definition symbolically evaluates to the area, and it, too, gets the right answer.

Anyone have any clue as to why the symbolic integration of this Heaviside function is so wrong? My understanding is that if we do the integral as two nested 1D integrals, the returned function (as a function of y) is zero everywhere except at y=0, but that result cannot be right either.

Thoughts?

 

Download Heaviside-error.mw

Greetings to all. The title describes it well, I am writing about testing the limits of the Maple integration engine. A recent discussion at math.stackexchange.com features a family of integrals that involve the product of a power of the natural logarithm and a rational function, more precisely,

int((log(x))^n/(x^3+1), x=0..infinity);

These integrals can be evaluated recursively as described at the MSE link using a technique that generalizes to other types of rational factors. Unfortunately Maple apparently only finds a simple closed form for a few small initial values of n. The following transcript of a Maple session illustrates the problem. Mathematica was successful here. Also observe the memory allocation in the Maple session.

    |\^/|     Maple 18 (X86 64 LINUX)
._|\|   |/|_. Copyright (c) Maplesoft, a division of Waterloo Maple Inc. 2014
 \  MAPLE  /  All rights reserved. Maple is a trademark of
 <____ ____>  Waterloo Maple Inc.
      |       Type ? for help.
> restart; read `cl.maple`;
alpha := (n, k) ->

                                                           n
    -1/3 exp(1/3 I Pi + 2/3 I Pi k) (1/3 I Pi + 2/3 I Pi k)

Q := proc(n)
local res;
option remember;
    if n = 0 then return 2/9*sqrt(3)*Pi end if;
    res := -add(alpha(n + 1, k), k = 0 .. 2)/(n + 1) - add(
        binomial(n + 1, p)*(2*I*Pi)^(n - p)*Q(p),
        p = 0 .. n - 1)/(n + 1);
    simplify(res)
end proc

                               infinity
                              /               n
                             |          log(x)
              VERIF := n ->  |          ------- dx
                             |           3
                            /           x  + 1
                              0

> Q(6);
                                7  1/2
                          910 Pi  3
                          ------------
                              6561

> VERIF(6);
memory used=3.8MB, alloc=40.3MB, time=0.18
       7  1/2
9890 Pi  3       490    5  1/2
------------- + ----- Pi  3    Psi(1, 1/3)
   177147       19683

        490    5  1/2                10    3  1/2            2
     + ----- Pi  3    Psi(1, 2/3) + ---- Pi  3    Psi(1, 1/3)
       19683                        2187

        20   1/2   3
     + ---- 3    Pi  Psi(1, 2/3) Psi(1, 1/3)
       2187

        10    3  1/2            2    40                 4
     + ---- Pi  3    Psi(1, 2/3)  + ----- Psi(2, 2/3) Pi
       2187                         19683

        10   1/2               3
     + ---- 3    Pi Psi(1, 1/3)
       2187

       10               1/2               2
     + --- Psi(1, 2/3) 3    Pi Psi(1, 1/3)
       729

       10   1/2                           2
     + --- 3    Pi Psi(1, 1/3) Psi(1, 2/3)
       729

        10   1/2               3    40     4
     + ---- 3    Pi Psi(1, 2/3)  - ----- Pi  Psi(2, 1/3)
       2187                        19683

        20             2  1/2
     + ---- Psi(2, 2/3)  3    Pi
       6561

        40               1/2
     - ---- Psi(2, 2/3) 3    Psi(2, 1/3) Pi
       6561

        40    2
     + ---- Pi  Psi(2, 2/3) Psi(1, 1/3)
       2187

        40    2
     + ---- Pi  Psi(2, 2/3) Psi(1, 2/3)
       2187

        20   1/2            2
     + ---- 3    Psi(2, 1/3)  Pi
       6561

        40    2
     - ---- Pi  Psi(1, 1/3) Psi(2, 1/3)
       2187

        40    2
     - ---- Pi  Psi(1, 2/3) Psi(2, 1/3)
       2187

> evalf(Q(6));
                          725.5729634

> evalf(VERIF(6));
                          725.5729630

> quit
memory used=22.4MB, alloc=44.3MB, time=0.47
user@host:~/complex-logint$ math
Mathematica 10.0 for Linux x86 (64-bit)
Copyright 1988-2014 Wolfram Research, Inc.

In[1]:= Integrate[Log[z]^6/(1+z^3), {z, 0, Infinity}]

                7
          910 Pi
Out[1]= ------------
        2187 Sqrt[3]

In[2]:= N[Out[1]]

Out[2]= 725.573

In[3]:=
user@host:~/complex-logint$

My question for you all is what the appropriate techniques would be to get Maple to at least simplify the rather involved output from the integration engine to obtain a match of the closed form from the recursive equation.

Best regards, Marko Riedel.

cl-maple.txt

Hello Mapleprime users

I am having an issue with a numerical integration calculation. I have a large(ish) polynomial integrand and need to apply two integrations which I am using the _cuhre method for. See attached the minimum working example file.

Numerical_integration_HF.mw

Firstly some simplifications are done to the integrand and then a basic for loop which calculates the integration for r from 0 to 20 in steps of 0.1. The issue occurs when the loop hits r~2.5 (on my machine, Maple 2015, i5, 16GB ram). Up to that point the calculation is steady with the following stats per point calculated:

memory used=87.59MiB, alloc change=0 bytes, cpu time=3.45s, real time=3.15s, gc time=439.16ms


Then when the calculation gets to ~2.5 it just hangs and will not calculate past it. Any ideas as to what is going on here?

Any help would be appreciated with this issue.

Thank you in advance

Yeti

 

 

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