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