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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.

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

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  • $\begingroup$ Okay. So now how do I proceed with this? $\endgroup$ – Nina Ali Jul 23 '15 at 13:48
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$$\text{Mass}=\int_{\theta=0}^{\pi/2}\int_{\phi=0}^{\pi}\int_0^1r^2\cos^2\phi ~r^2\sin\phi \,dr\,d\phi\, d\theta$$

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