Triple Integrals Help

Question

Suppose $E$ is the sphere $x^2 + y^2 + z^2 = 1$ whose density at each point is proportional to the distance from the origin. Find an expression for the mass of $E$ as a Triple Integral and explain why it's difficult to compute

I believe it is difficult to compute because the region is a sphere and not a box but I'm not exactly sure how to write the triple integral

Answer 1

Hint:

If $\delta=kr$ is the density at distance $r$ from the center, than the mass of a spherical shell from $r$ and $r+dr$ is $m=\delta 4 \pi r^2 dr$.

Can you do from this?

Answer 2

For an arbitrary density $\rho$, the mass is expressible as a triple integral in spherical polar coordinates, viz. $\int_0^{2\pi}d\phi\int_0^\pi d\theta\sin\theta\int_0^1 \rho(r,\,\theta,\,\phi)r^2 dr$. If $\rho$ only depends on $r$ we can first integrate out the angles, giving $4\pi\int_0^1\rho(r)r^2 dr$. The choice $\rho=kr$ from your question gives $4\pi k\int_0^1 r^3 dr=\pi k$.