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
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