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A sphere B have radius R. The center point is located the origin of space $\mathbf R^3$. The density of a sphere depends on the distance $r=\sqrt{x^2+y^2+z^2}$.

I want to calculate the mass of that sphere.

mass $m = \iiint \rho\, dV$ = $\int_V\rho(r)\, dV$

The volume $V(r) = {4 \over 3} \pi r^3 \to {d V(r) \over dr} = 4 \pi r^2$

After that we get:

$4\pi\int\rho(r)*r^2\,dr$

But what are the limit values of the integration?

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The distance to the origin can range from $r=0$ (at the origin) to $r=R$ (on the surface). Hence the bounds of integration are $0$ and $R$; in other words, your integral is $$4\pi\int_0^R\rho(r)r^2\;dr.$$

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