Question

find the volume using method of cylindrical shells?

Answer 1

If it is a body of revolution, i.e. the graph of a function that is rotated around the x-axis, you can use the following formula: \[V=\pi \int\limits_{a}^{b}(f(x))^2dx\]You can still see the cylinders there: radius is f(x), "height" = dx, so the volume of a single (infinitesimally thin) cylinder is π*(f(x))²*dx. The integral just adds them all up from x=a to x=b.
Now if you have a function f(x), just calculate the square of it and look for the primitive function (could be hard...)

Answer 2

ZeHanz, although this interpretation isn't exactly wrong, it's just not the shell method, it's the disk method because it gives you a bunch of disks that are all the same width, dx.

The shell method is really using a bunch of hollow shells to form the volume around the y-axis much like a bunch of tin cans like this:



See how the radius of the can is x, the height is f(x)? The integral for shell method is then:
\[2\pi \int\limits_{a}^{b}xf(x)dx\]

because the formula for surface area of a cylinder is \[A=2\pi r h\] which is just circumference times the height.

Slightly different, but I feel like this more accurately describes a cylinder since these actually vary in height while in the disk method the only thing that varies from slice to slice is the radius.

Answer 3

thankx