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## Homework Statement

Find the volume of the solid bouded by the hyperboloid.

(x^2/a^2)+(y^2/b^2)-(z^2/c^2)=1

and the planes z=0 an z=h, h>0.

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- Thread starter xiaobai5883
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- #1

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Find the volume of the solid bouded by the hyperboloid.

(x^2/a^2)+(y^2/b^2)-(z^2/c^2)=1

and the planes z=0 an z=h, h>0.

- #2

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tried? If you have tried, what have you tried?

i've tried...

but it's a failure...

i tried to use the integradation by pi*(integration of the certain area)^2...

but failed because the ellipse is in circle form only can use that...

so I'm wrong...

what i get after asking my lecturer is...

use the coin theory or disk theory...

it is something that say cut the object into small small pieces...

and just add them up...

i totally know the idea but don't know how to start or where to start...

- #4

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err..

you mean f(z)=pi*a*b??

but this is the area of each ellipse only right??

how about the whole volume??

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Another way is see that, for any point (x,y) in the xy-plane, [itex]z= \pm\c\sqrt{1- x^2- y^2}[/itex] so you can integrate the difference between those (the height of a thin rectangle) over the ellipse [itex]x^2/a^2+ y^2/b^2= 1[/itex].

- #7

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Another way is see that, for any point (x,y) in the xy-plane, [itex]z= \pm\c\sqrt{1- x^2- y^2}[/itex] so you can integrate the difference between those (the height of a thin rectangle) over the ellipse [itex]x^2/a^2+ y^2/b^2= 1[/itex].

okok...

but still two things that i don't understand..

by integrating that area...

do i get the volume??

and the second part is the another thing that i don't understand...

and i thought should use the Reimann Sum or what what method to solve...

am i wrong??/

- #8

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In the cylinders case the area of every slice was constant and therefore independent of the integration variable. In your case however if you slice the hyperboloid up every slice will have a semi major/minor axis that depends on z. Let's call this function A(z), then the volume of the hyperboloid is simply adding all the slices between z=0 and z=h together, [itex]\int_0^h A(z)dz[/itex]. Now it is up to you to find A(z). The easiest way to envision how you do that is to draw a picture of the x-z plane and the y-z plane.

- #9

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In the cylinders case the area of every slice was constant and therefore independent of the integration variable. In your case however if you slice the hyperboloid up every slice will have a semi major/minor axis that depends on z. Let's call this function A(z), then the volume of the hyperboloid is simply adding all the slices between z=0 and z=h together, [itex]\int_0^h A(z)dz[/itex]. Now it is up to you to find A(z). The easiest way to envision how you do that is to draw a picture of the x-z plane and the y-z plane.

okok..

i understand already...

but got one question appear again..

is there any theory that mention we integrate a line equation will become area...

then we integrate a area will become volume...

i'm sure there should be a theory like that...

but i want the name...

because i want to know more about that..

thanks...

- #10

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Green's and Gauss' theorems may be what you're looking for.

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