Given the value $K$ multiplied by radius $r$, I can determine the amount of mass of the solid red sphere at the center of the image using the following: $K * r = mass$.
However, $K$ begins to diminish as $r$ increases past the radius of the solid sphere.
As such, how do I calculate the amount of mass between any two regions between $r1$ and $r2$? I assume it would be an integral with the following form $\int_{r1}^{r2} K(r) * r \,dr$, but I could be mistaken.
But more importantly, in a unified expression, how do I calculate the total amount of mass in both the solid red sphere plus the amount of mass in the diffused area up to some radius greater than the radius of the solid red sphere?
Thank you.
As you have defined it, the mass interior to $r$ is $Kr$. You are correct that if $K$ varies you need to integrate to get the mass in a spherical shell. Presumably that integral starts from $r$, so the mass interior to $r2$ would be $$Kr+\int_r^{r2}K(r)r\;dr$$ There is nothing wrong with having the integral if the radius is greater than $r$ and not if it isn't.