How to change the limits of integration
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I am attempting to solve the integral of the following...
$$int_{0}^{2 pi}int_{0}^{infty}e^{-r^2}rdrTheta $$
So I do the following step...
$$=2 piint_{0}^{infty}e^{-r^2}rdr$$
but then the next step is to substitute $s = -r^2$ which results in...
$$=2 piint_{- infty}^{0}frac{1}{2}e^{s}ds$$
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
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add a comment |
$begingroup$
I am attempting to solve the integral of the following...
$$int_{0}^{2 pi}int_{0}^{infty}e^{-r^2}rdrTheta $$
So I do the following step...
$$=2 piint_{0}^{infty}e^{-r^2}rdr$$
but then the next step is to substitute $s = -r^2$ which results in...
$$=2 piint_{- infty}^{0}frac{1}{2}e^{s}ds$$
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
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1
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$$s = -r^{2} implies ds = -2r dr implies -ds/2 = rdr$$ and as $r to infty$, $s to - infty$ and as $r to 0$, $s to 0$.
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– Mattos
Apr 9 at 23:44
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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
add a comment |
$begingroup$
I am attempting to solve the integral of the following...
$$int_{0}^{2 pi}int_{0}^{infty}e^{-r^2}rdrTheta $$
So I do the following step...
$$=2 piint_{0}^{infty}e^{-r^2}rdr$$
but then the next step is to substitute $s = -r^2$ which results in...
$$=2 piint_{- infty}^{0}frac{1}{2}e^{s}ds$$
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
$endgroup$
I am attempting to solve the integral of the following...
$$int_{0}^{2 pi}int_{0}^{infty}e^{-r^2}rdrTheta $$
So I do the following step...
$$=2 piint_{0}^{infty}e^{-r^2}rdr$$
but then the next step is to substitute $s = -r^2$ which results in...
$$=2 piint_{- infty}^{0}frac{1}{2}e^{s}ds$$
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
calculus integration limits
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
add a comment |
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
add a comment |
2 Answers
2
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oldest
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$s=-r^{2}$ gives $ds=-2rdr$ so $dr =-frac 1 {2r} ds$. Also, as $r$ increases from $0$ to $infty$, $s$ decreases from $0$ to $-infty$.
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1
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It should be noted that the minus sign from the substitution is then used to reverse the order of the limits.
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– John Doe
Apr 9 at 23:45
1
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@JohnDoe Right. I didn't mention it explicitly but that is what I meant.
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– Kavi Rama Murthy
Apr 9 at 23:49
add a comment |
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Do you really need substitution. We already know the antiderivative of $re^{-r^2}$ and it is $-e^{-r^2}over 2$
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2 Answers
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2 Answers
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$begingroup$
$s=-r^{2}$ gives $ds=-2rdr$ so $dr =-frac 1 {2r} ds$. Also, as $r$ increases from $0$ to $infty$, $s$ decreases from $0$ to $-infty$.
$endgroup$
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
add a comment |
$begingroup$
$s=-r^{2}$ gives $ds=-2rdr$ so $dr =-frac 1 {2r} ds$. Also, as $r$ increases from $0$ to $infty$, $s$ decreases from $0$ to $-infty$.
$endgroup$
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
add a comment |
$begingroup$
$s=-r^{2}$ gives $ds=-2rdr$ so $dr =-frac 1 {2r} ds$. Also, as $r$ increases from $0$ to $infty$, $s$ decreases from $0$ to $-infty$.
$endgroup$
$s=-r^{2}$ gives $ds=-2rdr$ so $dr =-frac 1 {2r} ds$. Also, as $r$ increases from $0$ to $infty$, $s$ decreases from $0$ to $-infty$.
answered Apr 9 at 23:44
Kavi Rama MurthyKavi Rama Murthy
74.9k53270
74.9k53270
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
add a comment |
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
add a comment |
$begingroup$
Do you really need substitution. We already know the antiderivative of $re^{-r^2}$ and it is $-e^{-r^2}over 2$
$endgroup$
add a comment |
$begingroup$
Do you really need substitution. We already know the antiderivative of $re^{-r^2}$ and it is $-e^{-r^2}over 2$
$endgroup$
add a comment |
$begingroup$
Do you really need substitution. We already know the antiderivative of $re^{-r^2}$ and it is $-e^{-r^2}over 2$
$endgroup$
Do you really need substitution. We already know the antiderivative of $re^{-r^2}$ and it is $-e^{-r^2}over 2$
edited Apr 9 at 23:47
answered Apr 9 at 23:45
HAMIDINE SOUMAREHAMIDINE SOUMARE
2,403215
2,403215
add a comment |
add a comment |
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$begingroup$
$$s = -r^{2} implies ds = -2r dr implies -ds/2 = rdr$$ 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