Evaluating Lebesgue integral on the unit ball

Question

Does anyone can help me to calculate a Lebesgue integral with the help of the following definition?

$\int f d \mu = \int_0^{\infty} \mu(f^{-1}((t,\infty]))dt \in [0,\infty]$.

The integral to calculate is the following: $\int_{B_1(0)} \vert x \vert^s d \mu(x)$ where $\mu$ denotes the n-dimensional Lebesgue measure and $B_1(0)$ the unit ball and $s \in \mathbb{R}$.

Is it possible here to work with the definition above?

Answer 1

Ok I hope that I am now on a right way:

In order to usw the definition of the Lebesgue integral above I determined $f^{-1}((t,+\infty])$ as $\{ x \in B_1(0) : \lvert x\rvert^s > t\}$. But in a next step, how can I get the measure of this set, i.e. $\mu(\{ x \in B_1(0) : \lvert x\rvert^s > t\})$?

Any suggestions :)?