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 Posted: Wed Dec 13, 2017 10:34 pm

Joined: Wed Nov 15, 2017 12:37 pm
Posts: 20
Can you help me calculate the mass of the solid bounded by the surfaces $x^2+y^2=2y$ and $z=\sqrt{x^2+y^2}$ given its density function $m(x,y,z)=\sqrt{x^2+y^2}$ ?

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 Posted: Fri Dec 15, 2017 6:32 am
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Joined: Mon Nov 09, 2015 1:36 am
Posts: 460
Location: Ioannina, Greece
andrew.tzeva wrote:
...the solid bounded by the surfaces $x^2+y^2=2y$ and $z=\sqrt{x^2+y^2}$ ...

The solid isn't well defined. The surface $x^2+y^2=2y$ is a cylinder and the surface $z=\sqrt{x^2+y^2}$ is the "upper" half of the double cone $z^2=x^2+y^2$. These two surfaces does not enclose a solid. (See figure)
Attachment:

On the other hand, the double cone $z^2=x^2+y^2$ and the cylinder $x^2+y^2=2y$ enclose a solid. Maybe this is the case...

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Grigorios Kostakos

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 Posted: Sun Dec 17, 2017 12:27 pm

Joined: Wed Nov 15, 2017 12:37 pm
Posts: 20
Is there a general rule to solve this kind of problems, given two surfaces that bound a solid and its density function ?

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 Posted: Sun Dec 17, 2017 3:51 pm
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Joined: Mon Nov 09, 2015 1:36 am
Posts: 460
Location: Ioannina, Greece
andrew.tzeva wrote:
Is there a general rule to solve this kind of problems, given two surfaces that bound a solid and its density function ?
Yes, there is a general formula to calculate the mass of a solid $S$ with given density function $f:S\subset\mathbb{R}^3\longrightarrow\mathbb{R}$ :
$\displaystyle\mathop{\iiint}\limits_{S}{f\, dS}\,.$

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Grigorios Kostakos

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 Posted: Mon Dec 18, 2017 2:29 pm

Joined: Wed Nov 15, 2017 12:37 pm
Posts: 20
Thank you!

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