Thread: volume as solid of revolution

1. volume as solid of revolution

Ok, I'm having a hard time understand this problem or what to do even.

Two cones of height 2 and base circle of radius 4 are stuck together at their bases, and a cylindrical section of radius 2 is cut along the axis of the resulting object. Evaluate an integral to find the volume as a solid of revolution.

Help on this would be much appreciated!

2. Originally Posted by intervade Ok, I'm having a hard time understand this problem or what to do even.

Two cones of height 2 and base circle of radius 4 are stuck together at their bases, and a cylindrical section of radius 2 is cut along the axis of the resulting object. Evaluate an integral to find the volume as a solid of revolution.

Help on this would be much appreciated!
from your description, here is a "side" view ... rotate the two red functions about the x-axis. the green line represents the side of the cylinder, whose volume will have to be subtracted.

3. Originally Posted by intervade Ok, I'm having a hard time understand this problem or what to do even.

Two cones of height 2 and base circle of radius 4 are stuck together at their bases, and a cylindrical section of radius 2 is cut along the axis of the resulting object. Evaluate an integral to find the volume as a solid of revolution.

Help on this would be much appreciated!
If you rotate the region bounded by the y-axis, y=2, and y=4-2x about the x-axis, you will get half of the object.

I didn't see skeeter's post. If you turn his cone on its side you would get the one I described.

skeeter, how did you make your drawing?

4. Originally Posted by intervade Ok, I'm having a hard time understand this problem or what to do even.

Two cones of height 2 and base circle of radius 4 are stuck together at their bases, and a cylindrical section of radius 2 is cut along the axis of the resulting object. Evaluate an integral to find the volume as a solid of revolution.

Help on this would be much appreciated!

you could allow the y-axis to be the line passing through the vertices of both cones.
One cone is above the x-axis, resting on it, and the other is below the x-axis,
attached to the top cone (they are joined together).

You can set up the situation using the following sketch

5. Ok, so just checking my work here, is this correct:

Sorry,

2[ Pi * integral[from 0 - 2][ (4-2x)^2 - 2^2 ]dx ] ?

6. Originally Posted by intervade Ok, so just checking my work here, is this correct:

Sorry,

2[ Pi * integral[from 0 - 2][ (4-2x)^2 - 2^2 ]dx ] ?
y=4-2x and y=2 intrsect at (1, 2) so your limits of integation are from 0 to 1

As skeeter pointed out, this object would be missing the tips of the cone.

7. Originally Posted by intervade Ok, so just checking my work here, is this correct:

Sorry,

2[ Pi * integral[from 0 - 2][ (4-2x)^2 - 2^2 ]dx ] ?
with limits from 0 to 1, that would be the volume of the center part rotated about the x-axis minus the cylinder's volume ...

now the big question is, should the ends be included or not?

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