# Find Total Mass of a Solid

I need help finding total mass of a solid. So this solid is defined by inequalities:

$$x^2+y^2+z^2\le 1, x\ge 0, y\ge 0$$ and has a mass density of $z^2$.

Maybe we can convert this to cylindrical. I'm not sure how to solve this.

• can you find a parametrization of the solid? – Surb Jul 23 '15 at 13:49
• @Surb No, I'm not sure how to do it in this case? – Nina Ali Jul 23 '15 at 13:51
• ok. Do you know what does the volume $\{(x,y,z)\mid x^2+y^2+z^2\leq 1\}$ represent? – Surb Jul 23 '15 at 13:54
• @surb. Yes it's the volume of a sphere. – Nina Ali Jul 23 '15 at 13:55
• actually it is the unit ball :) (the sphere is for $x^2+y^2+z^2{\color{red}=}1$), and do you know how to parametrize this unit ball? have a look here – Surb Jul 23 '15 at 13:56

## 2 Answers

Hint: Let $E$ denote the region of interest. Then the total mass of $E$ is $\iiint_E z^2\text{d}V$. Since $E$ is part of a ball of radius $1$ centered at the origin, it is best to use spherical coordinates.

• Okay. So now how do I proceed with this? – Nina Ali Jul 23 '15 at 13:48

$$\text{Mass}=\int_{\theta=0}^{\pi/2}\int_{\phi=0}^{\pi}\int_0^1r^2\cos^2\phi ~r^2\sin\phi \,dr\,d\phi\, d\theta$$