# volume as solid of revolution

• Feb 21st 2010, 01:01 PM
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!
• Feb 21st 2010, 01:15 PM
skeeter
Quote:

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.
• Feb 21st 2010, 01:17 PM
ione
Quote:

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?
• Feb 21st 2010, 01:24 PM
Quote:

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
• Feb 21st 2010, 02:54 PM
Ok, so just checking my work here, is this correct:

Sorry,

2[ Pi * integral[from 0 - 2][ (4-2x)^2 - 2^2 ]dx ] ?
• Feb 21st 2010, 03:14 PM
ione
Quote:

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.
• Feb 21st 2010, 03:24 PM
skeeter
Quote: