How to change the limits of integration The 2019 Stack Overflow Developer Survey Results Are In Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Integration limits of the double integral after conversion to the polar coordinatesConvergent or Divergent using LimitsHow do I solve a double integral with an absolute value?How are limits of integration changed?Converting limits of integrationHow to set the limits for Jacobian IntegrationLimits of integration in multivariable integrals during change of variablesChange of limits of integrationWhat theorem(s) is(are) used to change between the various improper integrals.how does an integral becoming negative effect limits of integration?

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How to change the limits of integration



The 2019 Stack Overflow Developer Survey Results Are In
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Integration limits of the double integral after conversion to the polar coordinatesConvergent or Divergent using LimitsHow do I solve a double integral with an absolute value?How are limits of integration changed?Converting limits of integrationHow to set the limits for Jacobian IntegrationLimits of integration in multivariable integrals during change of variablesChange of limits of integrationWhat theorem(s) is(are) used to change between the various improper integrals.how does an integral becoming negative effect limits of integration?










5












$begingroup$


I am attempting to solve the integral of the following...



$$int_0^2 piint_0^inftye^-r^2rdrTheta $$



So I do the following step...



$$=2 piint_0^inftye^-r^2rdr$$



but then the next step is to substitute $s = -r^2$ which results in...



$$=2 piint_- infty^0frac12e^sds$$



The limits of integration are reversed now and the $r$ somehow results in $1/2$.



Can someone explain why this works? Why did substituting cause the limits change and result in the integration above?










share|cite|improve this question









$endgroup$







  • 1




    $begingroup$
    $$s = -r^2 implies ds = -2r dr implies -ds/2 = rdr$$ and as $r to infty$, $s to - infty$ and as $r to 0$, $s to 0$.
    $endgroup$
    – Mattos
    Apr 9 at 23:44











  • $begingroup$
    This constellation of consequences is why I recommend to my students to set $s = r^2$. This gives fewer minus signs and since the substituent is monotonically increasing, does not reverse the order of the bounds of integration.
    $endgroup$
    – Eric Towers
    Apr 10 at 4:30















5












$begingroup$


I am attempting to solve the integral of the following...



$$int_0^2 piint_0^inftye^-r^2rdrTheta $$



So I do the following step...



$$=2 piint_0^inftye^-r^2rdr$$



but then the next step is to substitute $s = -r^2$ which results in...



$$=2 piint_- infty^0frac12e^sds$$



The limits of integration are reversed now and the $r$ somehow results in $1/2$.



Can someone explain why this works? Why did substituting cause the limits change and result in the integration above?










share|cite|improve this question









$endgroup$







  • 1




    $begingroup$
    $$s = -r^2 implies ds = -2r dr implies -ds/2 = rdr$$ and as $r to infty$, $s to - infty$ and as $r to 0$, $s to 0$.
    $endgroup$
    – Mattos
    Apr 9 at 23:44











  • $begingroup$
    This constellation of consequences is why I recommend to my students to set $s = r^2$. This gives fewer minus signs and since the substituent is monotonically increasing, does not reverse the order of the bounds of integration.
    $endgroup$
    – Eric Towers
    Apr 10 at 4:30













5












5








5





$begingroup$


I am attempting to solve the integral of the following...



$$int_0^2 piint_0^inftye^-r^2rdrTheta $$



So I do the following step...



$$=2 piint_0^inftye^-r^2rdr$$



but then the next step is to substitute $s = -r^2$ which results in...



$$=2 piint_- infty^0frac12e^sds$$



The limits of integration are reversed now and the $r$ somehow results in $1/2$.



Can someone explain why this works? Why did substituting cause the limits change and result in the integration above?










share|cite|improve this question









$endgroup$




I am attempting to solve the integral of the following...



$$int_0^2 piint_0^inftye^-r^2rdrTheta $$



So I do the following step...



$$=2 piint_0^inftye^-r^2rdr$$



but then the next step is to substitute $s = -r^2$ which results in...



$$=2 piint_- infty^0frac12e^sds$$



The limits of integration are reversed now and the $r$ somehow results in $1/2$.



Can someone explain why this works? Why did substituting cause the limits change and result in the integration above?







calculus integration limits






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Apr 9 at 23:40









BolboaBolboa

408616




408616







  • 1




    $begingroup$
    $$s = -r^2 implies ds = -2r dr implies -ds/2 = rdr$$ and as $r to infty$, $s to - infty$ and as $r to 0$, $s to 0$.
    $endgroup$
    – Mattos
    Apr 9 at 23:44











  • $begingroup$
    This constellation of consequences is why I recommend to my students to set $s = r^2$. This gives fewer minus signs and since the substituent is monotonically increasing, does not reverse the order of the bounds of integration.
    $endgroup$
    – Eric Towers
    Apr 10 at 4:30












  • 1




    $begingroup$
    $$s = -r^2 implies ds = -2r dr implies -ds/2 = rdr$$ and as $r to infty$, $s to - infty$ and as $r to 0$, $s to 0$.
    $endgroup$
    – Mattos
    Apr 9 at 23:44











  • $begingroup$
    This constellation of consequences is why I recommend to my students to set $s = r^2$. This gives fewer minus signs and since the substituent is monotonically increasing, does not reverse the order of the bounds of integration.
    $endgroup$
    – Eric Towers
    Apr 10 at 4:30







1




1




$begingroup$
$$s = -r^2 implies ds = -2r dr implies -ds/2 = rdr$$ and as $r to infty$, $s to - infty$ and as $r to 0$, $s to 0$.
$endgroup$
– Mattos
Apr 9 at 23:44





$begingroup$
$$s = -r^2 implies ds = -2r dr implies -ds/2 = rdr$$ and as $r to infty$, $s to - infty$ and as $r to 0$, $s to 0$.
$endgroup$
– Mattos
Apr 9 at 23:44













$begingroup$
This constellation of consequences is why I recommend to my students to set $s = r^2$. This gives fewer minus signs and since the substituent is monotonically increasing, does not reverse the order of the bounds of integration.
$endgroup$
– Eric Towers
Apr 10 at 4:30




$begingroup$
This constellation of consequences is why I recommend to my students to set $s = r^2$. This gives fewer minus signs and since the substituent is monotonically increasing, does not reverse the order of the bounds of integration.
$endgroup$
– Eric Towers
Apr 10 at 4:30










2 Answers
2






active

oldest

votes


















8












$begingroup$

$s=-r^2$ gives $ds=-2rdr$ so $dr =-frac 1 2r ds$. Also, as $r$ increases from $0$ to $infty$, $s$ decreases from $0$ to $-infty$.






share|cite|improve this answer









$endgroup$








  • 1




    $begingroup$
    It should be noted that the minus sign from the substitution is then used to reverse the order of the limits.
    $endgroup$
    – John Doe
    Apr 9 at 23:45






  • 1




    $begingroup$
    @JohnDoe Right. I didn't mention it explicitly but that is what I meant.
    $endgroup$
    – Kavi Rama Murthy
    Apr 9 at 23:49


















0












$begingroup$

Do you really need substitution. We already know the antiderivative of $re^-r^2$ and it is $-e^-r^2over 2$






share|cite|improve this answer











$endgroup$













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    2 Answers
    2






    active

    oldest

    votes








    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    8












    $begingroup$

    $s=-r^2$ gives $ds=-2rdr$ so $dr =-frac 1 2r ds$. Also, as $r$ increases from $0$ to $infty$, $s$ decreases from $0$ to $-infty$.






    share|cite|improve this answer









    $endgroup$








    • 1




      $begingroup$
      It should be noted that the minus sign from the substitution is then used to reverse the order of the limits.
      $endgroup$
      – John Doe
      Apr 9 at 23:45






    • 1




      $begingroup$
      @JohnDoe Right. I didn't mention it explicitly but that is what I meant.
      $endgroup$
      – Kavi Rama Murthy
      Apr 9 at 23:49















    8












    $begingroup$

    $s=-r^2$ gives $ds=-2rdr$ so $dr =-frac 1 2r ds$. Also, as $r$ increases from $0$ to $infty$, $s$ decreases from $0$ to $-infty$.






    share|cite|improve this answer









    $endgroup$








    • 1




      $begingroup$
      It should be noted that the minus sign from the substitution is then used to reverse the order of the limits.
      $endgroup$
      – John Doe
      Apr 9 at 23:45






    • 1




      $begingroup$
      @JohnDoe Right. I didn't mention it explicitly but that is what I meant.
      $endgroup$
      – Kavi Rama Murthy
      Apr 9 at 23:49













    8












    8








    8





    $begingroup$

    $s=-r^2$ gives $ds=-2rdr$ so $dr =-frac 1 2r ds$. Also, as $r$ increases from $0$ to $infty$, $s$ decreases from $0$ to $-infty$.






    share|cite|improve this answer









    $endgroup$



    $s=-r^2$ gives $ds=-2rdr$ so $dr =-frac 1 2r ds$. Also, as $r$ increases from $0$ to $infty$, $s$ decreases from $0$ to $-infty$.







    share|cite|improve this answer












    share|cite|improve this answer



    share|cite|improve this answer










    answered Apr 9 at 23:44









    Kavi Rama MurthyKavi Rama Murthy

    74.6k53270




    74.6k53270







    • 1




      $begingroup$
      It should be noted that the minus sign from the substitution is then used to reverse the order of the limits.
      $endgroup$
      – John Doe
      Apr 9 at 23:45






    • 1




      $begingroup$
      @JohnDoe Right. I didn't mention it explicitly but that is what I meant.
      $endgroup$
      – Kavi Rama Murthy
      Apr 9 at 23:49












    • 1




      $begingroup$
      It should be noted that the minus sign from the substitution is then used to reverse the order of the limits.
      $endgroup$
      – John Doe
      Apr 9 at 23:45






    • 1




      $begingroup$
      @JohnDoe Right. I didn't mention it explicitly but that is what I meant.
      $endgroup$
      – Kavi Rama Murthy
      Apr 9 at 23:49







    1




    1




    $begingroup$
    It should be noted that the minus sign from the substitution is then used to reverse the order of the limits.
    $endgroup$
    – John Doe
    Apr 9 at 23:45




    $begingroup$
    It should be noted that the minus sign from the substitution is then used to reverse the order of the limits.
    $endgroup$
    – John Doe
    Apr 9 at 23:45




    1




    1




    $begingroup$
    @JohnDoe Right. I didn't mention it explicitly but that is what I meant.
    $endgroup$
    – Kavi Rama Murthy
    Apr 9 at 23:49




    $begingroup$
    @JohnDoe Right. I didn't mention it explicitly but that is what I meant.
    $endgroup$
    – Kavi Rama Murthy
    Apr 9 at 23:49











    0












    $begingroup$

    Do you really need substitution. We already know the antiderivative of $re^-r^2$ and it is $-e^-r^2over 2$






    share|cite|improve this answer











    $endgroup$

















      0












      $begingroup$

      Do you really need substitution. We already know the antiderivative of $re^-r^2$ and it is $-e^-r^2over 2$






      share|cite|improve this answer











      $endgroup$















        0












        0








        0





        $begingroup$

        Do you really need substitution. We already know the antiderivative of $re^-r^2$ and it is $-e^-r^2over 2$






        share|cite|improve this answer











        $endgroup$



        Do you really need substitution. We already know the antiderivative of $re^-r^2$ and it is $-e^-r^2over 2$







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Apr 9 at 23:47

























        answered Apr 9 at 23:45









        HAMIDINE SOUMAREHAMIDINE SOUMARE

        2,220212




        2,220212



























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