Question

(Calculus) - I've been trying to understand this for hours.
Can someone clearly explain the shell method in integration?

Where does the 2 pi come from? The area of the little rectangle times 2pi equals the circumference of the surface..?

And why do we just write "x"? Why is it just a random number in between 0 and the radius? Is it because it's an infinite amount of rectangles?

Any help will be appreciated.

Answer 1

Volume of Right Circular Cylinder: \(\pi r^{2}\;h\)
dVolume of Right Circular Cylinder: \(2\pi r\;h\;dr\)
dVolume of Right Circular Cylinder: \(\pi r^{2}\;dh\)

Answer 2

Yes, that did not satisfy my question, though.

Answer 3

Look more closely. Think hard on exactly what an integral represents and does.

\(2\pi r\) is the circumference of a circle.
\(h\) is the height of an open cylinder
\(2\pi rh\) is the surface area of an open cylinder.
Add up all the surface areas, \(\int stuff\;dr\), across all possible radii, and you have your volume.

\(\pi r^{2}\) is the area of a disc.
Add up all the surface areas, \(\int stuff\;dh\), across all possible heights, and you have your volume.

x and y are not random. They are the limits of the radius or the height. They can't get any smaller than zero.

Answer 4

Thanks, but I'm a bit confused because that is not the surface area of a cylinder. Well, I guess there's a difference between an open cylinder and a closed one, yes? The surface area is different..

Answer 5

I meant that is was not filled. It's just a hollow shell. Wait, isn't that why we call it he "shell method"? It IS the surface area of a cylinder - LOTS of them. More accurately, it might be called a "Representative Cylinder". Add up all the surface areas and get the volume. It's what integrals do.