Having a hard time solving this problem
- Thread starter Aventura
- Start date
- Joined
- Feb 4, 2004
- Messages
- 16,582
Would it be correct to guess that you're supposed to find the volume of the solid found by revolving the given region about the given axis? Or are you supposed to be finding the surface area via line integrals, etc?
When you reply, please include a clear listing of your thoughts and efforts so far. Thank you!
Sorry about the confusion, I am trying to find the volume of the solid that is generated. I am aware that that I need to find the outer radius and the inner radius, although i am having trouble with finding these. From what I worked I got that the outer radius is 1+e^y and the inner radius to be 1. When I set up the integral I get something along these lines
The integral from 0 to 1 pi[((1+e^y)^2)-((1)^2)]dy
Im not sure if i am even remotely close but any help is much appreciated
The integral from 0 to 1 pi[((1+e^y)^2)-((1)^2)]dy
Im not sure if i am even remotely close but any help is much appreciated
- Joined
- Feb 4, 2004
- Messages
- 16,582
I think your derivation was something along the lines of the following:
. . . . .converting outer-radius function: \(\displaystyle y\, =\, \ln(x)\, \implies\, x\, =\, e^y\)
. . . . .outer radius: \(\displaystyle R\, =\, e^y\, -\, (-1)\, =\, e^y\, +\, 1\)
. . . . .inner radius: \(\displaystyle r\, =\, 0\, -\, (-1)\, =\, 0\, +\, 1\, =\, 1\)
. . . . .do volume by discs: outer circle's area, less inner circle's area,
. . . . .multiplied by tiny widths delta-y = dy, and summing areas
. . . . .\(\displaystyle \displaystyle \sum\, \left(\pi\, R^2\, -\, \pi\, r^2\right)(\Delta\, y)\, \implies\, \pi\, \int_0^1\, \left[\left(e^y\, +\, 1\right)^2\, -\, 1\right]\, dy\)
Your integral looks good to me.
Last edited: