Evaluate $I = \int_{B_1^n(0)} (a_1x_1 + \cdots + a_nx_n)^{2/3}$ where $a_j \in \mathbb{R}$.
Here's where I'm at, following a hint:
Consider an orthonormal transformation $T$ with the first row equal to $(a_1/\alpha,\cdots,a_n/\alpha)$, where $\alpha = \sqrt{\sum a_j^2}$. Then, if $y = Tx$, $\alpha y_1 = \sum a_jx_j$. Now we have $I = \int _{B_1^n(0)}(\alpha y_1)^{2/3}$. But here's where I am not sure what to do with this $y_1$ to actually evaluate the integral? Why does $T$ need to be orthonormal, not just orthogonal?
Thanks