Volume Between A Circular Paraboloid and a Plane

In summary, the conversation involves finding the volume of a solid E bounded by two equations in cylindrical coordinates. The bounds on z are 3 and 6, while the bounds on r are 0 and √3. The bounds for θ are unknown, but the integral is performed in the order dz dr dθ. The calculated answer of 9*pi is questioned for its legitimacy and further details are requested.
  • #1

Homework Statement



Find the volume of the solid E bounded by z = 3+x2 +y2 and z = 6.

Homework Equations





The Attempt at a Solution


I'm going to use cylindrical coordinates. So, I have,

z = 3 + r2

Clearly, my bounds on z are 3 and 6. If I project the intersection of the paraboloid and the plan onto the xy-plane, I have 3 = r^2. My bounds on r are then 0 and ±√3, but I reject the negative boundary. I will use the bounds 0 and 2*pi for θ, though I don't know how these bounds are known to be appropriate. So, if I perform a triple integral of r in the order dz dr dθ, I get an answer of 9*pi.

Does that sound legitimate?
 
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  • #2
TranscendArcu said:
So, if I perform a triple integral of r in the order dz dr dθ, I get an answer of 9*pi.

Does that sound legitimate?

The result is not correct. Show your work in detail, please.

ehild
 

1. What is the formula for calculating the volume between a circular paraboloid and a plane?

The formula for calculating the volume between a circular paraboloid and a plane is V = (1/2)πr^2h, where r is the radius of the circular base of the paraboloid and h is the distance between the vertex of the paraboloid and the plane.

2. How do you determine if the volume between a circular paraboloid and a plane is positive or negative?

The volume between a circular paraboloid and a plane is positive if the plane intersects the paraboloid above the vertex, and negative if the plane intersects below the vertex.

3. Can the volume between a circular paraboloid and a plane be negative?

Yes, the volume can be negative if the plane intersects the paraboloid below the vertex, resulting in a negative volume.

4. How does the radius of the circular base of the paraboloid affect the volume between the paraboloid and a plane?

The radius of the circular base does not affect the volume between the paraboloid and a plane. The volume is only determined by the height of the paraboloid and the distance between the vertex and the plane.

5. Is there a real-life application of calculating the volume between a circular paraboloid and a plane?

Yes, this concept is used in engineering and architecture to calculate the volume of objects such as storage tanks, silos, and domes.

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