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Homework Help: Surface area from bands: Calculus

  1. Jan 25, 2014 #1
    1. The problem statement, all variables and given/known data

    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

    3. 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.
    Last edited: Jan 25, 2014
  2. jcsd
  3. Jan 25, 2014 #2


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    hi cathy! :smile:
    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 … ? :wink:
  4. Jan 25, 2014 #3
    hello! :)

    but looking at the picture, the band isnt the slice of a cone, is it?
  5. Jan 25, 2014 #4
    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.
  6. Jan 25, 2014 #5


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    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.
  7. Jan 25, 2014 #6


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    hello cathy! :smile:
    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)
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