Find Volume b/w 2 Surfaces: x, y, z Equations

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In summary, the question is to find the volume between the two surfaces z = 2x^2 + y^2 and z = 4 - y^2. The surfaces intersect at a circle with a radius of sqrt 2, and the bounds for x, y, and z are -sqrt 2 to sqrt 2, -sqrt 2-x^2 to sqrt 2-x^2, and 4-y^2 to 2x^2 + y^2, respectively. The suggested method is to integrate 1 dx dy dz and use cylindrical coordinates due to the circular symmetry.
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Kuma
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Homework Statement



Find the volume between the two surfaces

z = 2x^2 + y^2

z = 4 - y^2

Homework Equations





The Attempt at a Solution



Ok so i found out that the surfaces intersect at a circle. When i solved i got

x^2 + y^2 = 2, so the circle has a radius of sqrt 2.

So these are the bounds i got. X goes from - sqrt 2 to sqrt 2, y goes from - sqrt 2-x^2 to sqrt 2-x^2 and z goes from 4-y^2 to 2x^2 + y^2.

Is that right? If so, what is the equation I am integrating? Is it always integrating 1 dx dy dz?
 
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  • #2
That would certainly work. And what would be the bounds on the z-integral?


(Also, because of the circular symmetry, I would be inclined to use cylindrical coordinates- but you certainly can do it the way you suggest.)
 

FAQ: Find Volume b/w 2 Surfaces: x, y, z Equations

What is the concept of finding volume between two surfaces using x, y, z equations?

The concept of finding volume between two surfaces using x, y, z equations is based on the principle of integration in calculus. It involves calculating the integral of a function in three-dimensional space using the given equations for x, y, and z.

What are the steps involved in finding the volume between two surfaces using x, y, z equations?

The first step is to set up the given x, y, z equations as the boundaries for the integral. Then, use the appropriate integration method (such as double or triple integration) to calculate the volume between the two surfaces. Finally, evaluate the integral to obtain the final volume value.

What are the applications of finding volume between two surfaces using x, y, z equations?

This concept has many applications in physics and engineering, such as calculating the volume of a solid object, finding the mass of an object with varying density, and determining the center of mass for a three-dimensional object.

Are there any limitations or assumptions when using x, y, z equations to find volume between two surfaces?

Yes, there are some limitations and assumptions when using this method. One limitation is that the surfaces must be well-defined and have a continuous boundary. Additionally, this method assumes that the surfaces are not intersecting or overlapping with each other.

What are some tips for efficiently finding volume between two surfaces using x, y, z equations?

Some tips include breaking the volume into smaller, simpler shapes to make the integration process easier, using symmetry to reduce the number of integrations needed, and verifying the final answer by using alternative methods or approximations.

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