Why does the order of integration matter for the function $f(x,y)=y$?
1
$begingroup$
Consider the integrals:
$$
I=int_{0}^{sqrt{2}}int_{y^2}^{2}y dxdy \
I'=int_{y^2}^{2}int_{0}^{sqrt{2}}y dydx
$$
From what I understand, the order of integration does not matter. However, as can be easily shown by hand or a simple online integration software:
$$I=1 \ I'=2-y^2$$
Why are these not the same?
integration multivariable-calculus multiple-integral
asked Jan 20 at 9:45
Pancake_SenpaiPancake_Senpai
25116
$endgroup$
add a comment |
1
$begingroup$
Consider the integrals:
$$
I=int_{0}^{sqrt{2}}int_{y^2}^{2}y dxdy \
I'=int_{y^2}^{2}int_{0}^{sqrt{2}}y dydx
$$
From what I understand, the order of integration does not matter. However, as can be easily shown by hand or a simple online integration software:
$$I=1 \ I'=2-y^2$$
Why are these not the same?
integration multivariable-calculus multiple-integral
asked Jan 20 at 9:45
Pancake_SenpaiPancake_Senpai
25116
$endgroup$
2
$begingroup$
Draw a picture of the region you are integrating over. Then you have a sporting chance of understanding what you did wrong. A rule: only the limits of the inner integral may depend on the variable. The limits of the outer integral must be constants.
$endgroup$
– Jyrki Lahtonen
Jan 20 at 10:19
add a comment |
1
1
1
$begingroup$
Consider the integrals:
$$
I=int_{0}^{sqrt{2}}int_{y^2}^{2}y dxdy \
I'=int_{y^2}^{2}int_{0}^{sqrt{2}}y dydx
$$
From what I understand, the order of integration does not matter. However, as can be easily shown by hand or a simple online integration software:
$$I=1 \ I'=2-y^2$$
Why are these not the same?
integration multivariable-calculus multiple-integral
asked Jan 20 at 9:45
Pancake_SenpaiPancake_Senpai
25116
$endgroup$
Consider the integrals:
$$
I=int_{0}^{sqrt{2}}int_{y^2}^{2}y dxdy \
I'=int_{y^2}^{2}int_{0}^{sqrt{2}}y dydx
$$
From what I understand, the order of integration does not matter. However, as can be easily shown by hand or a simple online integration software:
$$I=1 \ I'=2-y^2$$
Why are these not the same?
integration multivariable-calculus multiple-integral
integration multivariable-calculus multiple-integral
asked Jan 20 at 9:45
Pancake_SenpaiPancake_Senpai
25116
asked Jan 20 at 9:45
Pancake_SenpaiPancake_Senpai
25116
asked Jan 20 at 9:45
Pancake_SenpaiPancake_Senpai
25116
asked Jan 20 at 9:45
Pancake_SenpaiPancake_Senpai
25116
asked Jan 20 at 9:45
Pancake_SenpaiPancake_Senpai
25116
25116
2
$begingroup$
Draw a picture of the region you are integrating over. Then you have a sporting chance of understanding what you did wrong. A rule: only the limits of the inner integral may depend on the variable. The limits of the outer integral must be constants.
$endgroup$
– Jyrki Lahtonen
Jan 20 at 10:19
add a comment |
2
$begingroup$
Draw a picture of the region you are integrating over. Then you have a sporting chance of understanding what you did wrong. A rule: only the limits of the inner integral may depend on the variable. The limits of the outer integral must be constants.
$endgroup$
– Jyrki Lahtonen
Jan 20 at 10:19
2
2
$begingroup$
Draw a picture of the region you are integrating over. Then you have a sporting chance of understanding what you did wrong. A rule: only the limits of the inner integral may depend on the variable. The limits of the outer integral must be constants.
$endgroup$
– Jyrki Lahtonen
Jan 20 at 10:19
$begingroup$
Draw a picture of the region you are integrating over. Then you have a sporting chance of understanding what you did wrong. A rule: only the limits of the inner integral may depend on the variable. The limits of the outer integral must be constants.
$endgroup$
– Jyrki Lahtonen
Jan 20 at 10:19
add a comment |
1 Answer
1
active
oldest
votes
5
$begingroup$
You can reverse the order but the limits of integration change:
$$int_{y=0}^{sqrt{2}}left(int_{x=y^2}^{2}y dxright)dy=int_{x=0}^{2}left(int_{y=0}^{sqrt{x}}y dyright)dx.$$
edited Jan 20 at 10:13
Jyrki Lahtonen
109k13169373
answered Jan 20 at 9:49
Robert ZRobert Z
98.3k1067139
$endgroup$
2
$begingroup$
I took the liberty of adding a picture of the region the integral is over. My experience is that students confused with iterated integrals too often fail to do that. Mind you, their confusion does not end there, but my thesis is that without the picture (at least in their minds) they have no chance. Anyway, if you object, just ping me, and I will remove it.
$endgroup$
– Jyrki Lahtonen
Jan 20 at 10:15
$begingroup$
@JyrkiLahtonen A picture here helps a lot. Thanks!
$endgroup$
– Robert Z
Jan 20 at 10:17
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
5
$begingroup$
You can reverse the order but the limits of integration change:
$$int_{y=0}^{sqrt{2}}left(int_{x=y^2}^{2}y dxright)dy=int_{x=0}^{2}left(int_{y=0}^{sqrt{x}}y dyright)dx.$$
edited Jan 20 at 10:13
Jyrki Lahtonen
109k13169373
answered Jan 20 at 9:49
Robert ZRobert Z
98.3k1067139
$endgroup$
2
$begingroup$
I took the liberty of adding a picture of the region the integral is over. My experience is that students confused with iterated integrals too often fail to do that. Mind you, their confusion does not end there, but my thesis is that without the picture (at least in their minds) they have no chance. Anyway, if you object, just ping me, and I will remove it.
$endgroup$
– Jyrki Lahtonen
Jan 20 at 10:15
$begingroup$
@JyrkiLahtonen A picture here helps a lot. Thanks!
$endgroup$
– Robert Z
Jan 20 at 10:17
add a comment |
5
$begingroup$
You can reverse the order but the limits of integration change:
$$int_{y=0}^{sqrt{2}}left(int_{x=y^2}^{2}y dxright)dy=int_{x=0}^{2}left(int_{y=0}^{sqrt{x}}y dyright)dx.$$
edited Jan 20 at 10:13
Jyrki Lahtonen
109k13169373
answered Jan 20 at 9:49
Robert ZRobert Z
98.3k1067139
$endgroup$
2
$begingroup$
I took the liberty of adding a picture of the region the integral is over. My experience is that students confused with iterated integrals too often fail to do that. Mind you, their confusion does not end there, but my thesis is that without the picture (at least in their minds) they have no chance. Anyway, if you object, just ping me, and I will remove it.
$endgroup$
– Jyrki Lahtonen
Jan 20 at 10:15
$begingroup$
@JyrkiLahtonen A picture here helps a lot. Thanks!
$endgroup$
– Robert Z
Jan 20 at 10:17
add a comment |
5
5
5
$begingroup$
You can reverse the order but the limits of integration change:
$$int_{y=0}^{sqrt{2}}left(int_{x=y^2}^{2}y dxright)dy=int_{x=0}^{2}left(int_{y=0}^{sqrt{x}}y dyright)dx.$$
edited Jan 20 at 10:13
Jyrki Lahtonen
109k13169373
answered Jan 20 at 9:49
Robert ZRobert Z
98.3k1067139
$endgroup$
You can reverse the order but the limits of integration change:
$$int_{y=0}^{sqrt{2}}left(int_{x=y^2}^{2}y dxright)dy=int_{x=0}^{2}left(int_{y=0}^{sqrt{x}}y dyright)dx.$$
edited Jan 20 at 10:13
Jyrki Lahtonen
109k13169373
answered Jan 20 at 9:49
Robert ZRobert Z
98.3k1067139
edited Jan 20 at 10:13
Jyrki Lahtonen
109k13169373
edited Jan 20 at 10:13
Jyrki Lahtonen
109k13169373
edited Jan 20 at 10:13
Jyrki Lahtonen
109k13169373
109k13169373
answered Jan 20 at 9:49
Robert ZRobert Z
98.3k1067139
answered Jan 20 at 9:49
Robert ZRobert Z
98.3k1067139
answered Jan 20 at 9:49
Robert ZRobert Z
98.3k1067139
98.3k1067139
2
$begingroup$
I took the liberty of adding a picture of the region the integral is over. My experience is that students confused with iterated integrals too often fail to do that. Mind you, their confusion does not end there, but my thesis is that without the picture (at least in their minds) they have no chance. Anyway, if you object, just ping me, and I will remove it.
$endgroup$
– Jyrki Lahtonen
Jan 20 at 10:15
$begingroup$
@JyrkiLahtonen A picture here helps a lot. Thanks!
$endgroup$
– Robert Z
Jan 20 at 10:17
add a comment |
2
$begingroup$
I took the liberty of adding a picture of the region the integral is over. My experience is that students confused with iterated integrals too often fail to do that. Mind you, their confusion does not end there, but my thesis is that without the picture (at least in their minds) they have no chance. Anyway, if you object, just ping me, and I will remove it.
$endgroup$
– Jyrki Lahtonen
Jan 20 at 10:15
$begingroup$
@JyrkiLahtonen A picture here helps a lot. Thanks!
$endgroup$
– Robert Z
Jan 20 at 10:17
2
2
$begingroup$
I took the liberty of adding a picture of the region the integral is over. My experience is that students confused with iterated integrals too often fail to do that. Mind you, their confusion does not end there, but my thesis is that without the picture (at least in their minds) they have no chance. Anyway, if you object, just ping me, and I will remove it.
$endgroup$
– Jyrki Lahtonen
Jan 20 at 10:15
$begingroup$
I took the liberty of adding a picture of the region the integral is over. My experience is that students confused with iterated integrals too often fail to do that. Mind you, their confusion does not end there, but my thesis is that without the picture (at least in their minds) they have no chance. Anyway, if you object, just ping me, and I will remove it.
$endgroup$
– Jyrki Lahtonen
Jan 20 at 10:15
$begingroup$
@JyrkiLahtonen A picture here helps a lot. Thanks!
$endgroup$
– Robert Z
Jan 20 at 10:17
$begingroup$
@JyrkiLahtonen A picture here helps a lot. Thanks!
$endgroup$
– Robert Z
Jan 20 at 10:17
add a comment |
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2
$begingroup$
Draw a picture of the region you are integrating over. Then you have a sporting chance of understanding what you did wrong. A rule: only the limits of the inner integral may depend on the variable. The limits of the outer integral must be constants.
$endgroup$
– Jyrki Lahtonen
Jan 20 at 10:19