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Let $alpha in mathbb{R}$ and $lambda_n$ the Lebesgue measure on $mathbb{R^n}$.

Define $f:mathbb{R^n}backslash(0)tomathbb{R}, f(x)=leftlVert xrightrVert^alpha$

For which $alpha$ is $f$ Lebesgue integrable on $B_1(0):=lbrace x in mathbb{R^n}:leftlVert xrightrVert leq 1 rbrace$

and for which $alpha$ on $mathbb{R^n}backslash B_1(0)$?

Also, how to compute $int_{B_1(0)}f dlambda_n$ and $int_{mathbb{R^n}backslash B_1(0)}f dlambda_n$ if $f$ is integrable?

I tried:

$f$ is Lebesgue integrable on $B_1(0) Leftrightarrowint_{B_1(0)}|f| dlambda_n<infty$.

So if $alpha<infty$, then $f(x)$ is Lebesgue integrable.

For computing the integrals, I thought about using the transformation formula:

$int_{B_1(0)}leftlVert xrightrVert^alpha dlambda_n=alpha cdot int_{0}^{1}r^n d lambda_r$

Here I don't know how to continue. Is this method correct or is there another way to do this?

Consider spherical shells $S_{a,b}:={xin{mathbb R}^n,|,aleq|x|leq b}$. One then has
$$int_{S_{a,b}}|x|^alpha {rm d}(x)=omega_{n-1}int_a^b r^alpha>r^{n+1}>dr ,$$
whereby $omega_{n-1}$ denotes the $(n-1)$-dimensional surface area of $S^{n-1}subset{mathbb R}^n$. Now see what happens when $ato0+$ or $btoinfty$ for various values of $n$ and $alpha$.