Volume between cylinder and ellipsoid

In summary: You will get something much simpler.In summary, the problem is to find the volume bounded by a cylinder and an ellipsoid. The surfaces are given by the equations x^2+y^2=4 and 4x^2+4y^2+z^2=64. To solve this, the intersection of both curves is reparameterized to be x=2s*cos(t), y=2s*sin(t), and z=-8(s^2-4). Using this reparameterization, the integral form S-M is found to be -3x^2-3y^2-z^2+60. After plugging in the reparameterization, the integral is simplified to become 32π, which represents
  • #1
ramb
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0

Homework Statement


I have two surfaces, a cylinder and an ellipsoid. I want to find the volume bounded by those two surfaces. The sufaces are:
S
[tex]x^{2}+y^{2}=4[/tex]

and
M
[tex]4x^{2}+4y^{2}+z^{2}=64[/tex]

Homework Equations



reparameterize it to get it in the form:

[tex]\int\int_{D}(S-M)dA[/tex]

If you do this and reperameterize it, you need to multiply it by the determenant of the x and y values.

The Attempt at a Solution



I reperameterized the intersection of both curves to be:

[tex]x=2s*cos(t)[/tex]
[tex]y=2s*sin(t)[/tex]
[tex]z=-8(s^{2}-4)[/tex]

so using that integral form S - M
I got:

[tex]\int\int_{D}-3x^{2}-3y^{2}-z^{2}+60 dA[/tex]

plugging in the reperamiterization I got:

=[tex]\int\int_{R}-3(2scos(t))^{2}-3(2ssin(t))^{2}+16s^{2}-4|X_{s}*Y_{t}-X_{t}*Y_{s}| dA[/tex]

=[tex]\int\int_{R}(4s^{2}-4)(4s) dA[/tex]

=[tex]16\int_0^1\int_0^{2\pi}(s^{3}-s) dtds[/tex]= [tex]32\pi[/tex]

now I'm not sure if what I've done was correct, but can someone help me to make sure if it is, whether it be by this method or another method.

thanks.
 
Last edited:
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  • #2
First, the "volume bounded by those two surfaces" isn't very clear. Do you mean the volume inside of both? I am guessing that is what you want. If so you are making the problem way too complicated. Draw a picture. The xy domain is the interior of a circle. Use cylindrical coordinates.
 

Related to Volume between cylinder and ellipsoid

1. What is the formula for calculating the volume between a cylinder and an ellipsoid?

The formula for calculating the volume between a cylinder and an ellipsoid is V = πh(R^2 - (R^2 - r^2)sqrt(1 - k^2) + (r^2/2)(1 - k^2)ln((1 + k)/(1 - k))), where h is the height of the cylinder, R is the radius of the cylinder, r is the radius of the ellipsoid, and k is the ratio of the ellipsoid's height to its width.

2. How is the volume between a cylinder and an ellipsoid different from the volume of a cylinder or an ellipsoid alone?

The volume between a cylinder and an ellipsoid is the volume of the space that is enclosed by both shapes. It is different from the volume of a cylinder or an ellipsoid alone because it takes into account the space that overlaps between the two shapes.

3. Can the volume between a cylinder and an ellipsoid be negative?

No, the volume between a cylinder and an ellipsoid cannot be negative. It represents the amount of space enclosed by the two shapes, which is always a positive value.

4. How can the volume between a cylinder and an ellipsoid be used in real life applications?

The volume between a cylinder and an ellipsoid can be used in various real life applications such as calculating the volume of liquid or gas in a storage tank with an ellipsoidal head, determining the volume of a water tank with a cylindrical base and an ellipsoidal top, or calculating the volume of a river or lake that has an elliptical cross section.

5. Is there a simpler way to calculate the volume between a cylinder and an ellipsoid?

There is no simpler way to calculate the volume between a cylinder and an ellipsoid as it involves complex mathematical equations and requires knowledge of the dimensions of both shapes. However, there are online calculators and software programs available that can help with the calculations.

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