# Finding mass of a sphere given density = $1-\rho^2$ and radius = 1

I'm currently working on what should be a relatively simple problem but I'm not sure if I'm right in my answer. This is for Calc3 homework that I'm just trying to get a grasp on the concepts before the final.

So the question is a sphere of radius 1 has a density of $1-\rho^2$ at a point distance $\rho$ from the center. How can I get the total mass of the sphere given this information?

I know $Mass = Volume \cdot Density$ and $Volume = r^3(4\pi/3)$ since $r = 1$

$Volume = \frac{4\pi}{3}$

So that leaves me with $m_{total} = \frac{4\pi}{3(1-\rho^2)}$

What should I be doing from here? Or am I even where I should be at this point in the problem?

• Consider the mass of a small spherical shell. – Element118 Nov 27 '15 at 7:24

Your formula of mass = volume $\times$ density needs to be a bit modified here since the density is non-uniform. Every bit of volume of the sphere has a different density so you have to integrate it appropriately as follows:

$$M=\int_0^1 \text{density}\cdot dV = \int_0^1 (1-r^2) \cdot dV$$

and we know that $V=\frac{4}{3}\pi r^3$ where $r$ is the radius of the sphere

So $dV=4\pi r^2dr$

Hence $$M=\int_0^1 (1-r^2) \cdot 4\pi r^2dr$$ $$=4\pi \int_0^1(r^2-r^4)dr$$ $$=4\pi(\frac{1}{3}-\frac{1}{5})=\frac{8\pi}{15}$$

• Very helpful, thank you. – J0hn Nov 27 '15 at 7:37
• You're welcome, – SchrodingersCat Nov 27 '15 at 7:37
• schrodinger's cat shouldn't we do $2\int_0^{2\pi}\int_0^1\int_0^{\sqrt{1-x^2-y^2}}(1-r^2)rdrd\theta$. Using the cylindrical cooedinates – Upstart Jul 7 '17 at 22:59
• @Upstart Why use cylindrical coordinates? All of a sudden while working with a sphere? – SchrodingersCat Jul 8 '17 at 4:31

You are wrong. The density is not uniform so you need to integrate the following:

$$\int_0^r(4\pi\rho^2)(1-\rho^2)d\rho$$