What is the volume enclosed by a plane $y = 23$ and a torus of radius $27$ and inner radius $4$
2
$begingroup$
I want to find the volume on the right of the plane (the smaller volume).
I think that the torus can be thought of as a volume of revolution of a circle about the z-axis, and that the bounds for integration can be set as y = 23 and y = 27 + 4 or 31. I'm not sure if that would work though.
The parameters of the torus are:
$x = (27+4cos v)cos u$
$y = (27+4cos v)sin u$
$z = 4sin v$
And $4$ is the radius of the tube.
Thank you
integration multiple-integral
asked Jan 9 at 11:16
mini edenmini eden
112
$endgroup$
|
show 2 more comments
2
$begingroup$
I want to find the volume on the right of the plane (the smaller volume).
I think that the torus can be thought of as a volume of revolution of a circle about the z-axis, and that the bounds for integration can be set as y = 23 and y = 27 + 4 or 31. I'm not sure if that would work though.
The parameters of the torus are:
$x = (27+4cos v)cos u$
$y = (27+4cos v)sin u$
$z = 4sin v$
And $4$ is the radius of the tube.
Thank you
integration multiple-integral
asked Jan 9 at 11:16
mini edenmini eden
112
$endgroup$
$begingroup$
Is the inner radius $4$ or $14$?
$endgroup$
– Shubham Johri
Jan 9 at 11:20
$begingroup$
The inner radius is 4, and by inner radius, I mean the radius of the cylinder, if the torus was stretched out.
$endgroup$
– mini eden
Jan 9 at 11:25
$begingroup$
Step 1: Parametrize the torus. Then things should take care of themselves.
$endgroup$
– B. Goddard
Jan 9 at 11:25
$begingroup$
I added the parameters to the torus in the question as requested.
$endgroup$
– mini eden
Jan 9 at 11:32
$begingroup$
For parameter values $u=v=0,$ you get $x = 31,$ $y=0,$ $z = 0.$ Now find the points closest to the origin; these all lie on a circle in the $z=0$ plane. None of this matches your figure. Even if we change $27$ to $25$ it doesn't match. So the first thing is to make very, very sure you know what problem you're trying to solve and construct an appropriate figure for it.
$endgroup$
– David K
Jan 9 at 17:03
|
show 2 more comments
2
2
2
1
$begingroup$
I want to find the volume on the right of the plane (the smaller volume).
I think that the torus can be thought of as a volume of revolution of a circle about the z-axis, and that the bounds for integration can be set as y = 23 and y = 27 + 4 or 31. I'm not sure if that would work though.
The parameters of the torus are:
$x = (27+4cos v)cos u$
$y = (27+4cos v)sin u$
$z = 4sin v$
And $4$ is the radius of the tube.
Thank you
integration multiple-integral
asked Jan 9 at 11:16
mini edenmini eden
112
$endgroup$
I want to find the volume on the right of the plane (the smaller volume).
I think that the torus can be thought of as a volume of revolution of a circle about the z-axis, and that the bounds for integration can be set as y = 23 and y = 27 + 4 or 31. I'm not sure if that would work though.
The parameters of the torus are:
$x = (27+4cos v)cos u$
$y = (27+4cos v)sin u$
$z = 4sin v$
And $4$ is the radius of the tube.
Thank you
integration multiple-integral
integration multiple-integral
asked Jan 9 at 11:16
mini edenmini eden
112
asked Jan 9 at 11:16
mini edenmini eden
112
edited Jan 11 at 5:21
mini eden
asked Jan 9 at 11:16
mini edenmini eden
112
asked Jan 9 at 11:16
mini edenmini eden
112
asked Jan 9 at 11:16
mini edenmini eden
112
112
$begingroup$
Is the inner radius $4$ or $14$?
$endgroup$
– Shubham Johri
Jan 9 at 11:20
$begingroup$
The inner radius is 4, and by inner radius, I mean the radius of the cylinder, if the torus was stretched out.
$endgroup$
– mini eden
Jan 9 at 11:25
$begingroup$
Step 1: Parametrize the torus. Then things should take care of themselves.
$endgroup$
– B. Goddard
Jan 9 at 11:25
$begingroup$
I added the parameters to the torus in the question as requested.
$endgroup$
– mini eden
Jan 9 at 11:32
$begingroup$
For parameter values $u=v=0,$ you get $x = 31,$ $y=0,$ $z = 0.$ Now find the points closest to the origin; these all lie on a circle in the $z=0$ plane. None of this matches your figure. Even if we change $27$ to $25$ it doesn't match. So the first thing is to make very, very sure you know what problem you're trying to solve and construct an appropriate figure for it.
$endgroup$
– David K
Jan 9 at 17:03
|
show 2 more comments
$begingroup$
Is the inner radius $4$ or $14$?
$endgroup$
– Shubham Johri
Jan 9 at 11:20
$begingroup$
The inner radius is 4, and by inner radius, I mean the radius of the cylinder, if the torus was stretched out.
$endgroup$
– mini eden
Jan 9 at 11:25
$begingroup$
Step 1: Parametrize the torus. Then things should take care of themselves.
$endgroup$
– B. Goddard
Jan 9 at 11:25
$begingroup$
I added the parameters to the torus in the question as requested.
$endgroup$
– mini eden
Jan 9 at 11:32
$begingroup$
For parameter values $u=v=0,$ you get $x = 31,$ $y=0,$ $z = 0.$ Now find the points closest to the origin; these all lie on a circle in the $z=0$ plane. None of this matches your figure. Even if we change $27$ to $25$ it doesn't match. So the first thing is to make very, very sure you know what problem you're trying to solve and construct an appropriate figure for it.
$endgroup$
– David K
Jan 9 at 17:03
$begingroup$
Is the inner radius $4$ or $14$?
$endgroup$
– Shubham Johri
Jan 9 at 11:20
$begingroup$
Is the inner radius $4$ or $14$?
$endgroup$
– Shubham Johri
Jan 9 at 11:20
$begingroup$
The inner radius is 4, and by inner radius, I mean the radius of the cylinder, if the torus was stretched out.
$endgroup$
– mini eden
Jan 9 at 11:25
$begingroup$
The inner radius is 4, and by inner radius, I mean the radius of the cylinder, if the torus was stretched out.
$endgroup$
– mini eden
Jan 9 at 11:25
$begingroup$
Step 1: Parametrize the torus. Then things should take care of themselves.
$endgroup$
– B. Goddard
Jan 9 at 11:25
$begingroup$
Step 1: Parametrize the torus. Then things should take care of themselves.
$endgroup$
– B. Goddard
Jan 9 at 11:25
$begingroup$
I added the parameters to the torus in the question as requested.
$endgroup$
– mini eden
Jan 9 at 11:32
$begingroup$
I added the parameters to the torus in the question as requested.
$endgroup$
– mini eden
Jan 9 at 11:32
$begingroup$
For parameter values $u=v=0,$ you get $x = 31,$ $y=0,$ $z = 0.$ Now find the points closest to the origin; these all lie on a circle in the $z=0$ plane. None of this matches your figure. Even if we change $27$ to $25$ it doesn't match. So the first thing is to make very, very sure you know what problem you're trying to solve and construct an appropriate figure for it.
$endgroup$
– David K
Jan 9 at 17:03
$begingroup$
For parameter values $u=v=0,$ you get $x = 31,$ $y=0,$ $z = 0.$ Now find the points closest to the origin; these all lie on a circle in the $z=0$ plane. None of this matches your figure. Even if we change $27$ to $25$ it doesn't match. So the first thing is to make very, very sure you know what problem you're trying to solve and construct an appropriate figure for it.
$endgroup$
– David K
Jan 9 at 17:03
|
show 2 more comments
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$begingroup$
Is the inner radius $4$ or $14$?
$endgroup$
– Shubham Johri
Jan 9 at 11:20
$begingroup$
The inner radius is 4, and by inner radius, I mean the radius of the cylinder, if the torus was stretched out.
$endgroup$
– mini eden
Jan 9 at 11:25
$begingroup$
Step 1: Parametrize the torus. Then things should take care of themselves.
$endgroup$
– B. Goddard
Jan 9 at 11:25
$begingroup$
I added the parameters to the torus in the question as requested.
$endgroup$
– mini eden
Jan 9 at 11:32
$begingroup$
For parameter values $u=v=0,$ you get $x = 31,$ $y=0,$ $z = 0.$ Now find the points closest to the origin; these all lie on a circle in the $z=0$ plane. None of this matches your figure. Even if we change $27$ to $25$ it doesn't match. So the first thing is to make very, very sure you know what problem you're trying to solve and construct an appropriate figure for it.
$endgroup$
– David K
Jan 9 at 17:03