# Volume Integral

• Oct 15th 2011, 08:04 AM
imagemania
Volume Integral
Question:
The volume V between
$\displaystyle z = \frac{5{x}^{2} + 3{y}^{2}}{R}$

And the x, y plane (z = xy?)

What is the volume of this region?

Thoughts?
$\displaystyle \iiint dV = \iiint dx dy dz$
Given that this is a paraboloid (3D) im not too sure how to approach it. I guess it just wants the volume of the paraboloid not entirely sure if the z = xy changes anything seeming the paraboloid doesn't go into negative z anyway?

Any help on how i should approach and deal with this is much appreciated!
• Oct 15th 2011, 08:21 AM
Prove It
Re: Volume Integral
Quote:

Originally Posted by imagemania
Question:
The volume V between
$\displaystyle z = \frac{5{x}^{2} + 3{y}^{2}}{R}$

And the x, y plane (z = xy?)

What is the volume of this region?

Thoughts?
$\displaystyle \iiint dV = \iiint dx dy dz$
Given that this is a paraboloid (3D) im not too sure how to approach it. I guess it just wants the volume of the paraboloid not entirely sure if the z = xy changes anything seeming the paraboloid doesn't go into negative z anyway?

Any help on how i should approach and deal with this is much appreciated!

What is R?
• Oct 15th 2011, 08:47 AM
DeMath
Re: Volume Integral
Maybe $\displaystyle z=x+y$ or $\displaystyle z=x-y$??
• Oct 15th 2011, 09:13 AM
imagemania
Re: Volume Integral
Sorry, R is a constant
• Oct 15th 2011, 12:54 PM
SammyS
Re: Volume Integral
The xy plane is the plane given by z = 0 .
• Oct 15th 2011, 02:58 PM
imagemania
Re: Volume Integral
Sorry im still not following,

I understand teh diagram looks of the order of this:
http://img202.imageshack.us/img202/4418/captureidmd.png

And this questions asks for the volume between that and the xy plane i.e. the infinite square that supports it.

Though i can't see converting the coordinate system as useful. I thought about doing:
$\displaystyle \vec{N} = r_{x} \times r_{y}$
Where:
$\displaystyle r_{x} = \frac{\partial x}{\partial x} + \frac{\partial y}{\partial x} ...$
but then N is a vector and not useful here as i want scalars!
• Oct 15th 2011, 03:33 PM
DeMath
Re: Volume Integral
Quote:

Originally Posted by imagemania
Question:
The volume V between
$\displaystyle z = \frac{5{x}^{2} + 3{y}^{2}}{R}$

And the x, y plane

What is the volume of this region?
Any help on how i should approach and deal with this is much appreciated!

Maybe the author of this problem implied such equations of planes: $\displaystyle z=x,~z=y$??

Then we can calculate the finite volume.
• Oct 16th 2011, 12:43 AM
imagemania
Re: Volume Integral
I'll word the question fully, V will be the volume
Consider V inside the cylinder
$\displaystyle x^{2} + y^{2} = 6{R}^{2}$
and between $\displaystyle z = (\frac{5{x}^{2} + 3{y}^{2}}{R})$ and the (x, y) plane.

x, y and z are the Cartesian coordinates and R is also a constant.

Now i read this question and thought it meant "Find the volume of the cylinder then find teh volume betwen the other two independently" i.e. two separate questions. Perhaps i was wrong?

Finding V for teh cylinder is just a matter of finding the jacobian and using cylindrical coordinates surely?

But the other part, assuming it is a separate question, i am not sure how to best approach it

Thanks
• Oct 16th 2011, 02:46 AM
DeMath
Re: Volume Integral
• Oct 18th 2011, 03:43 AM
imagemania
Re: Volume Integral
Oh i see now DeMath, it is simply using the first as z the and then introducing the cylindrical coordinates - didn't help i miss understood the question ;) thanks a lot :)

Though i must find out, how did you draw that image? I currently use Maple (14) and haven't figured how to draw them within the same body, what did you use and how did you do it?
• Oct 18th 2011, 11:26 AM
DeMath
Re: Volume Integral
Quote:

Originally Posted by imagemania
Though i must find out, how did you draw that image? I currently use Maple (14) and haven't figured how to draw them within the same body, what did you use and how did you do it?

You're lucky - this is done in Maple :)
Here is the code of this body (for $\displaystyle R=\sqrt{6}$)

plot3d([(5x^2+3y^2)/sqrt(6)], x=-6..6, y=-sqrt(36-x^2)..sqrt(36-x^2), filled=true, style=hidden, color="Cyan", lightmodel=light2, transparency=0.25, numpoints=10000, axes=normal, orientation=[81, 60])

When you see the picture, then move the cursor on her, click the right key of the mouse and after Esc.