Multivariable calculus involving considering symmetries and change of variables

Question

From a lesson that I had, there was this integral: (B denotes the unit ball)

$I_1=\iiint_{B} x^n+y^n+z^n dV = 3\iiint_B z^n dV$ and then the teacher used change of coordinates to spherical polar to deal with this integral and the rest is fairly standard. However I had a few questions. (Also the equality holds by symmetry, as stated by my teacher, if anyone is wondering)

$1)$ Say I do $3\iiint_B x^n dV$ instead and then use the spherical polar, would this give the same answer? (I believe it would because the unit ball is spherically symmetric)

$2)$ I later encounter a problem where I am required to evaluate $I_2=\iint_{D} (x+y)^{2n+1} dA$, where $D$ is the unit disc and I tried to do the same thing by saying $I_2=\iint_{D} (2y)^{2n+1} dA$ because I thought this is symmetric in a similar sense of the unit ball above. However my teacher said this approach is invalid. So why exactly is the first one valid and the second one isn't?

Answer 1

1. Yes. By symmetry, $\iiint_Bx^2\,\mathrm dV=\iiint_By^2\,\mathrm dV=\iiint_Bz^2\,\mathrm dV$.
2. Your teacher is right. You could do that if $n=0$, or you could say that the integral of $x^{2n+1}$ is equal to the integral of $y^{2n+1}$, but otherwise you are integrating a different function. Try to express exactly why you think that the integral of $(x+y)^{2n+1}$ is equal to the integral of $(2y)^{2n+1}$, using a symmetry principle. I suspect that you will not be able to do it.