prove integrability of $f(|x|)$ I would like some help on the following question.
$Let \ f:[0,1] \rightarrow \mathbb{R}$ be an integrable function.
prove that $f(|x|)$ is integrable in the closed - $k$ dimensional unit ball $B^k(0,1)$.
I tried using the Darboux sums but I could'nt find such partition that would make the Darboux sums of $f(|x|)$ converge.
 A: Now as you say you were trying to do by Darboux sums , I guess you were trying to prove it was Riemann integrable which is easy enough to do
Let $F_{n}:\Bbb{R}^{n}\to\Bbb{R}^{n}$ be the diffeomorphism
\begin{align}
x_1 &= r \cos(\varphi_1) \\
x_2 &= r \sin(\varphi_1) \cos(\varphi_2) \\
x_3 &= r \sin(\varphi_1) \sin(\varphi_2) \cos(\varphi_3) \\
    &\,\,\,\vdots\\
x_{n-1} &= r \sin(\varphi_1) \cdots \sin(\varphi_{n-2}) \cos(\varphi_{n-1}) \\
x_n     &= r \sin(\varphi_1) \cdots \sin(\varphi_{n-2}) \sin(\varphi_{n-1}) .
\end{align}
Then the Jacobian for the change of variables is given by $J_{n}=(r\sin(\varphi_1) \dotsm \sin(\varphi_{n-2}))|J_{n-1}|$ where $J_{n-1}$ is the jacobian for the diffeomorphism $F_{n-1}$.
Hence a simple induction shows that $|J_{n}|\leq 1 $ for all $n\in\Bbb{N}$ when $0\leq r\leq 1$.
The integral simply becomes :-
$$\int_{B[0,1]}f(|x|)\,dx = \int_{\varphi_{1},...,\varphi_{n-1}}\int_{0}^{1}f(r)|J_{n}|\,dr\,d\varphi_{1}...d\varphi_{n-1}\leq K\int_{0}^{1}f(r)\,dr<\infty$$ where $K$ is a finite positive constant determined by the ranges of $\varphi_{1},...,\varphi_{n-1}$ which are finite as the tuple $(\varphi_{1},...,\varphi_{n-1})$ must be coordinates for the sphere $S^{n-1}$.
However proving Lebesgue Integrability is a little more subtle affair
The following is from Stein and Shakarchi Real Analysis page 279.
Then the integral is:-
$$\int_{B[0,1]}f(|x|)\,d\lambda_{n}=\int_{S^{n-1}}\int_{\Bbb{R}}\mathbf{1}_{r\in[0,1]}f(r)\cdot r^{n-1}\,d\lambda\, d\mu$$ where $\mu$ is the measure on the sphere $S^{n-1}$ treated as a measure space whose definition I give below and $\lambda_{n}$ is the Lebesgue measure on $\Bbb{R}^{n}$.
Define a measure space $(X=S^{n-1},\mathcal{F},\mu)$ where $E\in\mathcal{F}$ if and only if the set $\hat{E}=\{x\in\Bbb{R}^{n} : \frac{x}{|x|}\in E\,,0<|x|<1\}$ is a Borel Measurable subset of $\Bbb{R}^{n}$ and $\mu(E)=n\cdot\lambda_{n}(\hat{E})$
Then by using the integrability of $f$ and the fact that $r\leq 1$ we have that by Fubini's Theorem :-
$$\int_{S^{n-1}}\int_{\Bbb{R}}\mathbf{1}_{r\in[0,1]}f(r)\cdot r^{n-1}\,d\lambda\, d\mu\leq n\lambda_{n}(B[0,1])\int_{[0,1]}f(r)\,d\lambda<\infty$$.
