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Centroid in three dimensions

  1. Nov 26, 2011 #1
    1. The problem statement, all variables and given/known data

    Find the centroid of the region D, lying above the sphere x2+y2+z2 and below the paraboloid z=4-x2-y2.

    2. Relevant equations

    3. The attempt at a solution

    I decided it might be easier to change it to polar coordinates soo I did
    M=∫∫∫ dzrdrdθ
    where z goes from sqrt(6-r2) to 4-r2
    θ goes from 0 to 2∏
    then i thought the r would be where the two functions intersect, so i set sqrt(6-r2)=4-r2 solving for r i got my solutions came out to be sqrt2 and sqrt5, i dont know which one to use because i thought r would go from 0 to either sqrt2 or sqrt5

    if some one could help me out i would really appreciate it.. thanks
    solving for r i got
  2. jcsd
  3. Nov 27, 2011 #2


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    Science Advisor

    Look at a graph. [itex]r^2+ z^2= 6[/itex] (you forgot the "6" in your origina equation but it can be deduced from what you wrote later) and [itex]z= 4- r^2[/itex] intersect twice. For [itex]r= \sqrt{2}[/itex], [itex]z= 4- 2= 2[/itex] and for [itex]r= \sqrt{5}[/itex], [itex]z= 4- 5= -1[/itex]. Because the problem says "above the sphere and below the paraoloid, you want to use the higher one, [itex]r= \sqrt{2}[/itex], z= 2.
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