1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Evalute surface area of top hemisphere

  1. Jan 6, 2012 #1

    sharks

    User Avatar
    Gold Member

    Evaluate surface area of top hemisphere

    The problem statement, all variables and given/known data
    http://s1.ipicture.ru/uploads/20120106/jrHc5122.jpg

    The attempt at a solution

    This problem is new to me, in the sense that the integration is to be done against S, which is the same region S, over which the limits are defined. Usually, the region and the d.w.r.to is different. For example, it is typical to use D as the underset and then dA instead of dS. dA is to find the area, but i would assume that dS means the same?

    The projection is done on the xy-plane so z = g(x,y) is the equation of the surface.
    So, making z the subject of formula:
    [tex]z=\sqrt{1-x^2-y^2}[/tex]
    [tex]f_x=\frac{\partial z}{\partial x}=\frac{-x}{\sqrt{1-x^2-y^2}}[/tex]
    [tex]f_y=\frac{\partial z}{\partial x}=\frac{-y}{\sqrt{1-x^2-y^2}}[/tex]

    [tex]SA=\underset{S}{{\iint}} \sqrt{1+(f_x)^2+(f_y)^2}\: dA[/tex]
    [tex]\sqrt{1+(f_x)^2+(f_y)^2} = \frac{1}{\sqrt{1-x^2-y^2}}[/tex]

    The boundaries for circular region S:
    For x fixed, y varies from y=0 to [itex]y=√(1−x^2)[/itex]
    x varies from x=0 to x=1

    Since it's a circular region, i should convert to polar coordinates for easier integration:

    For θ fixed, r varies from 0 to 1.
    θ varies from 0 to 2∏

    The integrand is converted to polar coordinates form:
    [tex]\frac{1}{\sqrt{1-x^2-y^2}}=\frac{1}{\sqrt{1-r^2}}[/tex]

    [tex]Surface\; area\;of\;S=\int_0^{2\pi}\int_0^1 \frac{1}{\sqrt{1-r^2}}\,.rdrd\theta[/tex]

    For some reason, my final integral above is wrong, as i'm supposed to get this instead:
    [tex]Surface\; area=\int_0^{2\pi}\int_0^1 r^2\,.rdrd\theta[/tex]

    I just followed the standard procedure for finding the surface area, but there is the integrand from the original equation itself that i have not included in that integral, [itex](x^2+y^2)z[/itex] and i have no idea how.

    I'm going to try and convert it to polar coordinates as well:
    [itex](x^2+y^2)z=r^2\sqrt{1-r^2}[/itex]

    By instinct, i'm just going to multiply this converted integrand with the other integrand, which gives me [itex]r^2[/itex] and this gives me the correct answer! Was that a fluke or did i follow the right steps?
     
    Last edited: Jan 6, 2012
  2. jcsd
  3. Jan 6, 2012 #2

    ehild

    User Avatar
    Homework Helper
    Gold Member

    It seems to me that you need to calculate the surface integral of f(x,y,z)=(x2+y2)z on the top semisphere instead of the surface area of the semisphere. Am I right?

    ehild
     
  4. Jan 6, 2012 #3

    sharks

    User Avatar
    Gold Member

    Hi ehild!

    I have revised my calculations in the first post.
     
    Last edited: Jan 6, 2012
  5. Jan 6, 2012 #4

    ehild

    User Avatar
    Homework Helper
    Gold Member

    It is correct now.

    ehild
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Evalute surface area of top hemisphere
Loading...