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Calculation of the mass of solid bounded by two surfaces

Multivariate Calculus
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andrew.tzeva
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Calculation of the mass of solid bounded by two surfaces

#1

Post by andrew.tzeva » Wed Dec 13, 2017 10:34 pm

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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Grigorios Kostakos
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Re: Calculation of the mass of solid bounded by two surfaces

#2

Post by Grigorios Kostakos » Fri Dec 15, 2017 6:32 am

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)
solidmass.png
solidmass.png (24.14 KiB) Viewed 1861 times
[/centre]
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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Re: Calculation of the mass of solid bounded by two surfaces

#3

Post by andrew.tzeva » Sun Dec 17, 2017 12:27 pm

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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Grigorios Kostakos
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Re: Calculation of the mass of solid bounded by two surfaces

#4

Post by Grigorios Kostakos » Sun Dec 17, 2017 3:51 pm

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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Re: Calculation of the mass of solid bounded by two surfaces

#5

Post by andrew.tzeva » Mon Dec 18, 2017 2:29 pm

Thank you!
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