Triple integral - switching limits around
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I have been attempting this question for more than four hours now. Since I don't have the answer key for it, I try and check my answer using Symbolab by ensuring all three integrals give the same volume. However, I haven't managed to get the that to happen till now.
For part (a): $z$ should range from $x^2 to 3-y$, $x$ should range from $-sqrt{3-y} to sqrt{3-y}$, and $y$ should range from $0 to 2$. Are these correct for this part?
Following the confirmation that my answer for part (a) is right:
For part (b): We can split it into two integrals:
$x$ should range from $-sqrt{z} to sqrt{z}$, $y$ should range from $0 to 3-z$ and $z$ should range from $1 to 3$.
$x$ should range from $-sqrt{z} to sqrt{z}$, $y$ should range from $0 to 2$ and $z$ should range from $0 to 1$.
Is this also correct?
integration multivariable-calculus definite-integrals volume
asked Dec 3 '18 at 21:27
Gummy bearsGummy bears
1,88811531
$endgroup$
add a comment |
$begingroup$
I have been attempting this question for more than four hours now. Since I don't have the answer key for it, I try and check my answer using Symbolab by ensuring all three integrals give the same volume. However, I haven't managed to get the that to happen till now.
For part (a): $z$ should range from $x^2 to 3-y$, $x$ should range from $-sqrt{3-y} to sqrt{3-y}$, and $y$ should range from $0 to 2$. Are these correct for this part?
Following the confirmation that my answer for part (a) is right:
For part (b): We can split it into two integrals:
$x$ should range from $-sqrt{z} to sqrt{z}$, $y$ should range from $0 to 3-z$ and $z$ should range from $1 to 3$.
$x$ should range from $-sqrt{z} to sqrt{z}$, $y$ should range from $0 to 2$ and $z$ should range from $0 to 1$.
Is this also correct?
integration multivariable-calculus definite-integrals volume
asked Dec 3 '18 at 21:27
Gummy bearsGummy bears
1,88811531
$endgroup$
$begingroup$
Only a little correction, $y$ varies from $0$ to $3$ as the intersection of the plane with the line $y=0$ is at $y=3$
$endgroup$
– Rafa Budría
Dec 3 '18 at 22:25
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@RafaBudría But the question says that there's the plane $y = 2$ to consider? Shouldn't we stop integrating at that point?
$endgroup$
– Gummy bears
Dec 3 '18 at 22:27
$begingroup$
You are right. Your solution is ok.
$endgroup$
– Rafa Budría
Dec 3 '18 at 22:30
$begingroup$
@RafaBudría I updated the answer. Could you please also check that?
$endgroup$
– Gummy bears
Dec 3 '18 at 22:42
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You might get some help by looking at my YouTube lecture on changing order of integration. Definitely the middle third of this one, and perhaps the previous lecture, too.
$endgroup$
– Ted Shifrin
Dec 3 '18 at 23:14
add a comment |
$begingroup$
I have been attempting this question for more than four hours now. Since I don't have the answer key for it, I try and check my answer using Symbolab by ensuring all three integrals give the same volume. However, I haven't managed to get the that to happen till now.
For part (a): $z$ should range from $x^2 to 3-y$, $x$ should range from $-sqrt{3-y} to sqrt{3-y}$, and $y$ should range from $0 to 2$. Are these correct for this part?
Following the confirmation that my answer for part (a) is right:
For part (b): We can split it into two integrals:
$x$ should range from $-sqrt{z} to sqrt{z}$, $y$ should range from $0 to 3-z$ and $z$ should range from $1 to 3$.
$x$ should range from $-sqrt{z} to sqrt{z}$, $y$ should range from $0 to 2$ and $z$ should range from $0 to 1$.
Is this also correct?
integration multivariable-calculus definite-integrals volume
asked Dec 3 '18 at 21:27
Gummy bearsGummy bears
1,88811531
$endgroup$
I have been attempting this question for more than four hours now. Since I don't have the answer key for it, I try and check my answer using Symbolab by ensuring all three integrals give the same volume. However, I haven't managed to get the that to happen till now.
For part (a): $z$ should range from $x^2 to 3-y$, $x$ should range from $-sqrt{3-y} to sqrt{3-y}$, and $y$ should range from $0 to 2$. Are these correct for this part?
Following the confirmation that my answer for part (a) is right:
For part (b): We can split it into two integrals:
$x$ should range from $-sqrt{z} to sqrt{z}$, $y$ should range from $0 to 3-z$ and $z$ should range from $1 to 3$.
$x$ should range from $-sqrt{z} to sqrt{z}$, $y$ should range from $0 to 2$ and $z$ should range from $0 to 1$.
Is this also correct?
integration multivariable-calculus definite-integrals volume
integration multivariable-calculus definite-integrals volume
asked Dec 3 '18 at 21:27
Gummy bearsGummy bears
1,88811531
asked Dec 3 '18 at 21:27
Gummy bearsGummy bears
1,88811531
edited Dec 3 '18 at 22:42
Gummy bears
asked Dec 3 '18 at 21:27
Gummy bearsGummy bears
1,88811531
asked Dec 3 '18 at 21:27
Gummy bearsGummy bears
1,88811531
asked Dec 3 '18 at 21:27
Gummy bearsGummy bears
1,88811531
1,88811531
$begingroup$
Only a little correction, $y$ varies from $0$ to $3$ as the intersection of the plane with the line $y=0$ is at $y=3$
$endgroup$
– Rafa Budría
Dec 3 '18 at 22:25
$begingroup$
@RafaBudría But the question says that there's the plane $y = 2$ to consider? Shouldn't we stop integrating at that point?
$endgroup$
– Gummy bears
Dec 3 '18 at 22:27
$begingroup$
You are right. Your solution is ok.
$endgroup$
– Rafa Budría
Dec 3 '18 at 22:30
$begingroup$
@RafaBudría I updated the answer. Could you please also check that?
$endgroup$
– Gummy bears
Dec 3 '18 at 22:42
$begingroup$
You might get some help by looking at my YouTube lecture on changing order of integration. Definitely the middle third of this one, and perhaps the previous lecture, too.
$endgroup$
– Ted Shifrin
Dec 3 '18 at 23:14
add a comment |
$begingroup$
Only a little correction, $y$ varies from $0$ to $3$ as the intersection of the plane with the line $y=0$ is at $y=3$
$endgroup$
– Rafa Budría
Dec 3 '18 at 22:25
$begingroup$
@RafaBudría But the question says that there's the plane $y = 2$ to consider? Shouldn't we stop integrating at that point?
$endgroup$
– Gummy bears
Dec 3 '18 at 22:27
$begingroup$
You are right. Your solution is ok.
$endgroup$
– Rafa Budría
Dec 3 '18 at 22:30
$begingroup$
@RafaBudría I updated the answer. Could you please also check that?
$endgroup$
– Gummy bears
Dec 3 '18 at 22:42
$begingroup$
You might get some help by looking at my YouTube lecture on changing order of integration. Definitely the middle third of this one, and perhaps the previous lecture, too.
$endgroup$
– Ted Shifrin
Dec 3 '18 at 23:14
$begingroup$
Only a little correction, $y$ varies from $0$ to $3$ as the intersection of the plane with the line $y=0$ is at $y=3$
$endgroup$
– Rafa Budría
Dec 3 '18 at 22:25
$begingroup$
Only a little correction, $y$ varies from $0$ to $3$ as the intersection of the plane with the line $y=0$ is at $y=3$
$endgroup$
– Rafa Budría
Dec 3 '18 at 22:25
$begingroup$
@RafaBudría But the question says that there's the plane $y = 2$ to consider? Shouldn't we stop integrating at that point?
$endgroup$
– Gummy bears
Dec 3 '18 at 22:27
$begingroup$
@RafaBudría But the question says that there's the plane $y = 2$ to consider? Shouldn't we stop integrating at that point?
$endgroup$
– Gummy bears
Dec 3 '18 at 22:27
$begingroup$
You are right. Your solution is ok.
$endgroup$
– Rafa Budría
Dec 3 '18 at 22:30
$begingroup$
You are right. Your solution is ok.
$endgroup$
– Rafa Budría
Dec 3 '18 at 22:30
$begingroup$
@RafaBudría I updated the answer. Could you please also check that?
$endgroup$
– Gummy bears
Dec 3 '18 at 22:42
$begingroup$
@RafaBudría I updated the answer. Could you please also check that?
$endgroup$
– Gummy bears
Dec 3 '18 at 22:42
$begingroup$
You might get some help by looking at my YouTube lecture on changing order of integration. Definitely the middle third of this one, and perhaps the previous lecture, too.
$endgroup$
– Ted Shifrin
Dec 3 '18 at 23:14
$begingroup$
You might get some help by looking at my YouTube lecture on changing order of integration. Definitely the middle third of this one, and perhaps the previous lecture, too.
$endgroup$
– Ted Shifrin
Dec 3 '18 at 23:14
add a comment |
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$begingroup$
Only a little correction, $y$ varies from $0$ to $3$ as the intersection of the plane with the line $y=0$ is at $y=3$
$endgroup$
– Rafa Budría
Dec 3 '18 at 22:25
$begingroup$
@RafaBudría But the question says that there's the plane $y = 2$ to consider? Shouldn't we stop integrating at that point?
$endgroup$
– Gummy bears
Dec 3 '18 at 22:27
$begingroup$
You are right. Your solution is ok.
$endgroup$
– Rafa Budría
Dec 3 '18 at 22:30
$begingroup$
@RafaBudría I updated the answer. Could you please also check that?
$endgroup$
– Gummy bears
Dec 3 '18 at 22:42
$begingroup$
You might get some help by looking at my YouTube lecture on changing order of integration. Definitely the middle third of this one, and perhaps the previous lecture, too.
$endgroup$
– Ted Shifrin
Dec 3 '18 at 23:14