proof of coarea formula for n dimensional hypersurface in $R^n$

Question

$f:R^n \rightarrow R$ be continuous and summable.

please give the proof for these formulas

$\int_{R^n}f dx = \int_0^\infty(\int_{\partial B(x_0,r)}fdS)dr$

$\frac{d}{dr}\int_{ B(x_0,r)}fdx=\int_{\partial B(x_0,r)}fdS$

simpler method would be better?

Answer 1

I don't have any knowledge of proving this for hypersurfaces.I've tried to give a non-rigorous answer to the second equation given.However I'm not very sure if my usage of the fundamental theorem of calculus is correct because $F$ is not defined at zero.
$$F(t):=\int_{dB(x,t)}f(y)dS(y) $$ $$\int_{B(x,r)}f(w)dw = \int_0^rF(t) dt$$ $${d\over dr}\int_{B(x,r)}f(w)dw ={d\over dr}\int_0^rF(t) dt = F(r) = \int_{dB(x,r )}f(y)dS(y)$$