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Centroid in three dimensions

  • Thread starter MozAngeles
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  • #1
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Homework Statement



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

Homework Equations





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
 

Answers and Replies

  • #2
HallsofIvy
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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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