|Nov11-07, 01:39 AM||#1|
1. The problem statement, all variables and given/known data
Let U be the solid region in the first octant bounded by the ellipsoid (x^2)/4 + (y^2)/9 + (z^2)/4 = 1. Find the centroid of U.
2. Relevant equations
3. The attempt at a solution
I tried to do this problem but I'm not sure if my answer is right. First, I find the mass and I got 2pi. Then I find the moment, and divided it by mass, and I got the centroid to be like (1,9/8,1). But I'm not sure if I did it right or not. Can anyone help me?
|Nov11-07, 07:14 AM||#2|
That can't be right. Since the "y-length" is 3, and the other lengths 2, the y coordinate of the centroid must be 3/2 the other coordinates.
I did this by changing to "elliptic coordinates". More precisely, I let
[itex]x= 2\rho cos(\theta) sin(\phi)[/itex], [itex]y= 3\rho sin(\theta) sin(\phi)[/itex] and [itex]z= 2\rho cos(\phi)[/itex]- spherical coordinates 'altered' to fit the ellipse.
Calculating the Jacobian gives [itex]12\rho^2 sin(\phi)d\rho d\phi d\theta[/itex] as the differential. The ellipse in the first octant take [itex]\rho[/itex] from 0 to 1, [itex]\phi[/itex] from 0 to [itex]\pi/2[/itex] and [itex]\theta[/itex] from 0 to [itex]\pi/2[/itex].
I get (3/8, 9/16, 3/8) as the centroid.
(By the way, [itex]2\pi[/itex] is the volume of the ellipse, not the "mass". "Centroid" is a purely geometrical concept and geometric figures do not have "mass".)
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