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!

Homework Help: Line integral to determine area of sphere?

  1. Oct 24, 2011 #1
    Find the area of the surface consisting of the part of the sphere of radius 2 centered at
    origin that lies above the horizontal plane z = 1. (Equation of this sphere is given by
    x^2 + y^2 + z^2 = 2^2 .)


    This is the base of the solid. But how do we find the required surface area of the sphere?
    How do we use line integral to determine this area? It's not an area that is parallel to the z-axis.
  2. jcsd
  3. Oct 24, 2011 #2


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Doesn't your text have a formula for the surface area element dS for z = f(x,y)?
    [tex]dS=\sqrt{1 + f_x^2+f_y^2}\ dxdy[/tex]
    or parametric versions for cylindrical and spherical coordinates?
  4. Oct 24, 2011 #3


    User Avatar
    Science Advisor

    I don't understand your remark about the base not being parallel to the z- axis. Of course, it isn't- it lies in the plane z= 1 and so is perpendicular to the z-axis.

    I would do this by writing parametric equations for the sphere using spherical coordinates. The surface can be written as [itex]x= 2cos(\theta)sin(\phi)[/itex], [itex]y= 2sin(\theta)sin(\phi)[/itex], [itex]z= 2cos(\phi)[/itex]. [itex]z= 2cos(\phi)= 1[/itex] gives [itex]cos(\phi)= 1/2[/itex] so [itex]\phi= \pi/3[/itex].

    The vector equation for that sphere would be [itex]\vec{r}(\theta, \phi)= 2cos(\theta)sin(\phi)\vec{i}+ 2sin(\theta)sin(\phi)\vec{j}+ 2cos(\phi)\vec{k}[/itex]. The derivative of that with respect to the two parameters gives two tangent vectors to the sphere. The cross product of those two vectors is then perpendicular to the sphere and its length, with [itex]d\theta d\phi[/itex] is the "differential of surface area". Integrate that with [itex]\theta[/itex] from 0 to [itex]2\pi[/itex] and [itex]\phi[/itex] from 0 to [itex]\pi/3[/itex].
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Similar Threads for Line integral determine
Line integral
Line integral of a curve
Line integral of a vector field