I'm having trouble with finding the volume of y=x^2 from x=0, to x=1, revolving about the x axis. I'm having trouble visualizing it with this method. I can execute it with other methods, but not shell.

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- February 24th 2011, 12:07 PMquantoembryoshell method
I'm having trouble with finding the volume of y=x^2 from x=0, to x=1, revolving about the x axis. I'm having trouble visualizing it with this method. I can execute it with other methods, but not shell.

- February 24th 2011, 01:16 PMArchie Meade
Using the washer method, it's easy to visualise them by comparing them to the rings of Saturn.

For the shells, pick a point on the y-axis at about 0.8

and draw a horizontal line until it touches the curve above the positive x-axis.

Rotate this line about the x-axis.

It traces out a cylinder.

You want the curved surface area of this cylinder (the surface area of the shell).

Now imagine doing the same for all the other horizontal lines from y=0 to y=1.

Integrate all the surface areas to get the volume (like layers of an onion). - February 24th 2011, 01:48 PMquantoembryo
When I envision what you say, it comes up wrong. The shell will have dimensions 2pi*f(x), x, and dxm I get dV=2pi*x^3dx which can't be right. Help?

- February 24th 2011, 02:16 PMArchie Meade
- February 24th 2011, 02:37 PMquantoembryo
I don't know what my issues are with this.... I can now see that it is dy, but that's about it. I keep ending up with 2pi*x^3 and I know that's wrong. My textbook gives one example and I just don't get it..

- February 24th 2011, 02:55 PMArchie Meade
- February 24th 2011, 03:17 PMquantoembryo
Alright, I integrated that and ended up with 4pi/5, however, I did it via splices and got pi/5 which I know is right

- February 24th 2011, 03:47 PMArchie Meade
- February 24th 2011, 06:11 PMquantoembryo
Thanks a lot for the help. However, my final concern is I don't see why the height is 1-x rather than just x, even from your diagram

- February 25th 2011, 02:30 AMArchie Meade
All of the cylinders have to rest against the line x=1.

This is the case when we are calculating the VOR of the region

between the curve and horizontal axis using shells.

I've included 2 such cylinders, though there are "infinitely" many.

Therefore we need to subtract x from 1 to get the cylinder heights,

since the lines of length 1-x are being rotated around the horizontal axis.

Hope this helps.

Remember, these cylinders are "resting on their sides", not standing vertically. - February 25th 2011, 04:51 AMquantoembryo
Yes, I finally am understanding. Thanks for your patience!

- February 25th 2011, 05:09 AMArchie Meade
Thanks, sorry for not distinguishing between the region above the curve under the imaginary line y=1

and the region below the curve above the horizontal axis earlier in the thread.