Finding a volume in space

In summary,The person seeking help is trying to find the volume between the plane z=1 and z=10-x*x-4*y*y. They have tried using polar coordinates, but they are stuck integrating the expression over an ellipse. They have also tried using cartesian coordinates, but they are also stuck. The person asks for help finding an elegant way to integrate the expression.
  • #1
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Hello, I need to find the volume between the plane z=1 and z=10-x*x-4*y*y.

I have tried using polar coordinates:
first integrate from z=1 to z=10-x*x-4*y*y
The I integrate in the plane z=1, but here I need to integrate over an ellipse, how do I do this? When I wrote the expression to polar coordinates I got an expression for r that I couldn't integrate.

I have also tried integrating over the ellipse using cartesian coordinates, but I got stuck aswell, can someone pleas help?
 
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  • #2
Could you show the actual function you have to integrate over? I suspect it has the form [itex]\sqrt{9-4y^2}[/itex]? If so use the substitution [itex]u=\frac{3}{2} \sin \theta[/itex]. Still integrating like this isn't very pretty. A more elegant method and by far the easiest to integrate is to notice that this is a paraboloid with elliptical cross section. You can then slice the paraboloid parallel to the x-y plane in a lot of slices and then integrate from z=1 to 10. The only thing you have to do is find an expression for the semi major axis and the semi minor axis as a function of z and use that the area of an ellipse is given by pi a b, with a the semi major axis and b the semi minor axis.
 
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  • #3
Cyosis said:
Could you show the actual function you have to integrate over? I suspect it has the form [itex]\sqrt{9-4y^2}[/itex]? If so use the substitution [itex]u=\frac{3}{2} \sin \theta[/itex]. Still integrating like this isn't very pretty. A more elegant method and by far the easiest to integrate is to notice that this is a paraboloid with elliptical cross section. You can then slice the paraboloid parallel to the x-y plane in a lot of slices and then integrate from z=1 to 10. The only thing you have to do is find an expression for the semi major axis and the semi minor axis as a function of z and use that the area of an ellipse is given by pi a b, with a the semi major axis and b the semi minor axis.
No, finding the volume bounded by z= 1 and [itex]z= 10- x^2- 4y^2[/itex] involves integrating the difference in z values, [itex](10- x^2- 4y^2)- 1= 9- x^2- y^2[/tex] over the circle [itex]x^2+ 4y^2= 9[/itex], where the two surfaces cross.

Now the limits of integration, in xy-coordinates will involve something like [itex]y= \frac{1}{2}\sqrt{9- x^2}[/itex].

Hyper, you might consider using parameters r and [itex]\theta[/itex] such that [itex]x= r cos(\theta)[/itex] and [itex]y= 4r sin(\theta)[/itex], not standard polar coordinates, with r going from 0 to 1 and [itex]\theta[/itex] from 0 to [itex]2\pi[/itex]. Do you know how to find the "differential of area" in r and [itex]\theta[/itex] using the Jacobian?
 
  • #4
Why wouldn't slices be applicable, it yields the same answer?
 

1. How do you calculate the volume of an object in space?

The volume of an object in space can be calculated by multiplying its length, width, and height. However, in order to find the volume of an irregularly shaped object, one would need to use mathematical techniques such as integration or approximation.

2. What is the unit of measurement for volume in space?

The standard unit of measurement for volume is cubic meters (m3) in space. Other commonly used units include cubic centimeters (cm3) and liters (L).

3. Can the volume of an object in space change?

Yes, the volume of an object in space can change due to various factors such as compression, expansion, or chemical reactions. For example, a gas can expand and occupy a larger volume when heated.

4. How is the volume of a gas in space measured?

The volume of a gas in space is typically measured using its pressure and temperature. This is known as the Ideal Gas Law, which states that the volume of a gas is directly proportional to its temperature and inversely proportional to its pressure.

5. Can the volume of an object in space be negative?

No, the volume of an object in space cannot be negative as it represents the amount of space an object occupies. However, the volume of a gas can be negative if it is under extreme pressure and occupying a smaller space than its ideal volume at standard pressure and temperature.

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