# Rotating circle

• Nov 17th 2006, 11:54 AM
Jones
Rotating circle
Hi, i need some help on this.

a) Determine the volume dV of the cylindric(?) shell in the figure.

the curve is y=16-x^2

I took a picture of the image in the book, might help

http://img452.imageshack.us/my.php?i...ture022oj5.jpg
• Nov 17th 2006, 12:29 PM
CaptainBlack
Quote:

Originally Posted by Jones
Hi, i need some help on this.

a) Determine the volume dV of the cylindric(?) shell in the figure.

the curve is y=16-x^2

I took a picture of the image in the book, might help

ImageShack - Hosting :: picture022oj5.jpg

$\displaystyle dV=(2\ \pi\ x)\ y\ dx=(2\ \pi\ x)\ (16 - x^2)\ dx$

RonL
• Nov 17th 2006, 12:37 PM
Jones
Quote:

Originally Posted by CaptainBlack
$\displaystyle dV=(2\ \pi\ x)\ y\ dx=(2\ \pi\ x)\ (16 - x^2)\ dx$

RonL

Hm, could you explain how you did that?
• Nov 17th 2006, 12:41 PM
CaptainBlack
Quote:

Originally Posted by Jones
Hm, could you explain how you did that?

The circumference of the shell is $\displaystyle 2 \pi\ x$ the thickness is
$\displaystyle dx$ the height is $\displaystyle y$. To first order in $\displaystyle dx$ the volume is:

$\displaystyle dV= (2 \pi x) y dx$

If you want it more explicit its the difference in volume of the two cylinders of radius $\displaystyle x$
and $\displaystyle x+dx$ of heights $\displaystyle y(x)$ and $\displaystyle y(x+dx)$ or:

$\displaystyle dV= -\pi x^2 y(x) + \pi (x+dx)^2 y(x+dx)=\pi\ 2 \ x\ dx (16-x^2) +O(dx^2)$

RonL