Original post by grhas98
Hi,
In this question I’m not sure where I am going wrong, I am wondering if I am doing it wrong altogether, can you just integrate from r= 0 to r= R and multiply by 2pi?
In this question I’m not sure where I am going wrong, I am wondering if I am doing it wrong altogether, can you just integrate from r= 0 to r= R and multiply by 2pi?
It looks as though what you've done is to find the area under the curve from r = 0 to r = R and multiplied it by 2pi, whereas what the question is asking for is a volume of rotation about the y (height) axis. There's more than one way of finding the volume of rotation, but what I'd suggest in this case is to consider the overall 3D shape as being made up of a series of concentric cylindrical shells, and integrate accordingly.
(edited 10 months ago)
Original post by old_engineer
It looks as though what you've done is to find the area under the curve from r = 0 to r = R and multiplied it by 2pi, whereas what the question is asking for is a volume of rotation about the y (height) axis. There's more than one way of finding the volume of rotation, but what I'd suggest in this case is to consider the overall 3D shape as being made up of a series of concentric cylindrical shells, and integrate accordingly.
Ah so use cylindrical coordinates and dV = rdrdzdtheta?
Original post by grhas98
Ah so use cylindrical coordinates and dV = rdrdzdtheta?
No not really. My thinking was that if the volume of a vertical cylindrical shell can be represented by 2pi(r)(h)(dr) then you can find the volume of the object as a whole by integrating that with respect to r between r = 0 and r = R. That will effectively give you the sum of the volumes of all the concentric cylindrical shells between the central vertical axis and the outer limits of the shape.
Original post by mqb2766
You just need a sketch and use
h = Re^(-r/R)
So get r(h) and integrate the volume of a thin disc pi*r^2*dh over the appropriate limits.
h = Re^(-r/R)
So get r(h) and integrate the volume of a thin disc pi*r^2*dh over the appropriate limits.
That will doubtless work, but (a) it does involved integrating a squared ln() function, and (b) you have to deal separately with the fixed radius cylinder forming the base of the 3D shape.
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