Triple integral/spherical coordinates (ellipsoid)

[Melissa00]

Hi

I'm having some issues with this math problem:
∫_E z² dxdydz with the ellipsoid x²/4+y²/9+z² ≤ 1

So I thought that the first thing I should do is transforming the actual equation into spherical coordinates (?)
z² --> r sinθ [from lecture notes]

How am I going to determine the limits for the integrals for dr, and dφ dθ?
For φ I thought 0 ≤ φ ≤ 2π; though I'm not even sure about this.

I'm sure that this is easy to solve, but I already looked though tons of material (online and also books)... I'm seriously having some problems understanding this.

I'd greatly appreciate your help

 

[DrPhil]

Hi

I'm having some issues with this math problem:
∫_E z² dxdydz with the ellipsoid x²/4+y²/9+z² ≤ 1

So I thought that the first thing I should do is transforming the actual equation into spherical coordinates (?)
z² --> r sinθ [from lecture notes]

How am I going to determine the limits for the integrals for dr, and dφ dθ?
For φ I thought 0 ≤ φ ≤ 2π; though I'm not even sure about this.

I'm sure that this is easy to solve, but I already looked though tons of material (online and also books)... I'm seriously having some problems understanding this.

I'd greatly appreciate your help

Since all three axes of the ellipsoid are different, it isn't clear to me that finding the limits of integration in spherical coordinates would be any easier than in Cartesian coordinates. Since the ellipsoid is centered at the origin, we can integrate just the first octant and multiply by 8:

\(\displaystyle \displaystyle 8 \int_0^{??} \left( \int_0^{??} \left( \int_0^{??} z^2\ dz \right) dy \right) dx \)

To find the limits, start with the outermost integration. What is the upper limit of x?
Then your upper limit for y will be a function of x,
and your upper limit for z will be a function of both x and y.

Take another try at it, and show us how far you can get.

EDIT: taking 8 times the integral in one octant also requires that the integrand by symmetric on all 3 axes - which it is.

 

[DrPhil]

∫_E z² dxdydz with the ellipsoid x²/4+y²/9+z² ≤ 1

A substitution that will make the problem easier to write:

Let u = x/2,......dx = 2 du
......v = y/3,......dy = 3 dv

Then the formula for the ellipsoid becomes a unit sphere in the (u,v,z) coordinate system:

......\(\displaystyle \displaystyle r^2 = u^2 + v^2 + z^2 \leq 1\)


......\(\displaystyle \displaystyle 8 \int_0^{??} \left( \int_0^{??} \left( \int_0^{??} z^2\ dz \right) 3\ dv \right) 2\ du\)

The effect of the substitution has been to multiply the integral by 6. You may proceed in Cartesian coordinates, OR convert to spherical if you prefer. BTW, don't forget the square on z^2 when converting.

 

[Melissa00]

Thanks

Thank you for you suggestions, but unfortunately out professor wanted us to use the spherical coordinates to solve this :| Which is way too complicated in this case, because as your already mentioned, there are easier ways to solve this.

I'll post the suggested answer tonight,...

Thanks again