triple integral... volume

[mathstresser]

Evaluate \(\displaystyle \L\\\int_{m}^{m}\int_{m}^{m}\int_{m}^{m} (x^2)(y)dv\), where is the solid of:

The ellipsoid
$\displaystyle (x^2)/(a^2) + (y^2)/(b^2) + (z^2)/(c^2) = 1.$

Use the transformation
x=au
y=bv
z=cw

I get the Jacobian factor is abc.

Also I think the intervals for x,y, and z are
0<=x<=a
0<=y<=b
0<=z<=c

But I don’t know what else to do in the transformation… What intervals to evaluate the new integrals from.

Also- isn’t the new integral

\(\displaystyle \L\\\int_{m}^{m}\int_{m}^{m}\int_{m}^{m} a^3b^2cu^2vdudvdw\)

which is also

$\displaystyle a^3b^2c$\(\displaystyle \L\\\int_{m}^{m}int_{m}^{m}int_{m}^{m}u^2vdudvdw\)

Is that right?

Could somebody please help me?

(Sorry about all the random m's. I couldn't get it to look right w/o having variables in there!)


18