Evaluating the Integral: \iiint_{V} (16x^2 + 9y^2 + 4z^2)^{1/4} \,dx\,dy\,dz

[Stevecgz]

I am trying to evaluate the following:

$$\iiint_{V} (16x^2 + 9y^2 + 4z^2)^{1/4} \,dx\,dy\,dz$$

Where V is the ellipsoid $$16x^2 + 9y^2 + 4z^2 \leq 16$$

This is what I've done:

Change of variables with
$$u^2 = 16x^2$$
$$v^2 = 9y^2$$
$$w^2 = 4z^2$$

Then V is the sphere
$$u^2 + v^2 + z^2 \leq 16$$

And the jacobian is
$$\frac{1}{24}$$

Than another Change of variables to Spherical cordinates, so the resulting integral is:

$$\int_{0}^{2\pi} \int_{0}^{pi} \int_{0}^{4} (\rho^2)^{1/4}\rho^2\sin\phi\frac{1}{24} \,d\rho\, d\phi\, d\theta$$

My question is if I am going about this the correct way and if it is ok to make two change of variables as I have done. Thanks.

Steve

 

[Gagle The Terrible]

You can always do as many change of variables as you wish, but be carefull with the jacobian. You can even "invent" your own set of coordinates and integrate in these particular coordinates.

 

[Stevecgz]

Thanks Gagle.