Find Centroid of U: First Octant Ellipsoid

[Jimmy21]

Homework Statement

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.

Homework Equations

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?

 

[HallsofIvy]

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
$x= 2\rho cos(\theta) sin(\phi)$, $y= 3\rho sin(\theta) sin(\phi)$ and $z= 2\rho cos(\phi)$- spherical coordinates 'altered' to fit the ellipse.
Calculating the Jacobian gives $12\rho^2 sin(\phi)d\rho d\phi d\theta$ as the differential. The ellipse in the first octant take $\rho$ from 0 to 1, $\phi$ from 0 to $\pi/2$ and $\theta$ from 0 to $\pi/2$.
I get (3/8, 9/16, 3/8) as the centroid.

(By the way, $2\pi$ is the volume of the ellipse, not the "mass". "Centroid" is a purely geometrical concept and geometric figures do not have "mass".)