# Computing the Lebesgue integral over a ball

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

Define $$f:\mathbb{R^n}\backslash(0)\to\mathbb{R}, \ f(x)=\left\lVert x\right\rVert^\alpha$$

For which $$\alpha$$ is $$f$$ Lebesgue integrable on $$B_1(0):=\lbrace x \in \mathbb{R^n}:\left\lVert x\right\rVert \leq 1 \rbrace$$

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

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

I tried:

$$f$$ is Lebesgue integrable on $$B_1(0) \Leftrightarrow\int_{B_1(0)}|f| \ d\lambda_n<\infty$$.

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

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

$$\int_{B_1(0)}\left\lVert x\right\rVert^\alpha \ d\lambda_n=\alpha \cdot \int_{0}^{1}r^n \ d \lambda_r$$

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

Consider spherical shells $$S_{a,b}:=\{x\in{\mathbb R}^n\,|\,a\leq\|x\|\leq b\}$$. 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 $$a\to0+$$ or $$b\to\infty$$ for various values of $$n$$ and $$\alpha$$.