Surface area from bands: Calculus

[cathy]

Homework Statement

The shaded band shown here is cut from a sphere of radius Rby parallel planes hunits apart. Show that the surface area of the band is 2piRh.
The image is on this site: http://imgur.com/TCx1weD
http://imgur.com/TCx1weD

The Attempt at a Solution

How do I do this? I thought that it was given that the dS= 2pi*r dL, so since dL=h, it would simply be dS=2pi*rh, but then that doesn't make a lot of sense because I would have to take the integralto find x, but what are the points that I am taking the integral from? I am a bit confused. Please advise if you can.

Anytime I do this problem, I'm getting that ds= 2piR*dL, which is where I'm trying to get, but to find s, wouldn't you have to take the integral of that? This is where I'm confused.

 

[tiny-tim]

hi cathy!

The shaded band shown here is cut from a sphere of radius Rby parallel planes h units apart. Show that the surface area of the band is 2piRh.

always use the slicing method …

slice the band into tiny slices of height dh, and radius a function of h

then each slice will be very nearly a slice of a cone, and you can take its surface area to be that of a slice of a cone, which is … ? ​

 

[cathy]

hello! :)

but looking at the picture, the band isn't the slice of a cone, is it?

 

[cathy]

if you take the band, I automatically thing that it should be 2piR* the thickness, which in this case is h, so why do I need to slice?
Are there calculations necessary here?

Sorry, I am very confused as to to show the proof.

 

[vanhees71]

Perhaps I misunderstand something, but what's to be calculated seems to be part of a spherical shell. So you should parametrize the sphere (hint: spherical coordinates with fixed radius are the natural choice) and think about where the parameters run to cover the piece of the shell you want to calculate.

 

[tiny-tim]

hello cathy!

but looking at the picture, the band isn't the slice of a cone, is it?

the bit of the Earth that you're living on is part of a slice of the same latitude, λ, that goes all the way round the earth

you probably think it looks flat!

so you'd calculate its area as the area of a slice of a cone at angle λ

(and the reason why you don't use dh is because the surface is slanting … dh is the difference in height, but the actual distance from top to bottom is longer)