Finding volume bounded by paraboloid and cylinder

In summary: Variable domain or constant domain?Variable domain or constant domain?I think the z1 function would be the function of the paraboloid, and the volume would be a subtraction of the cylinder function z2 from z1.
  • #1
iqjump123
61
0

Homework Statement


Find the volume bounded by the paraboloid z= 2x2+y2 and the cylinder z=4-y2. Diagram is included that shows the shapes overlaying one another, with coordinates at intersections. (Will be given if necessary)


Homework Equations


double integral? function1-function2?


The Attempt at a Solution


I saw from previous threads involving volumes, but still am lost when I try to do my own problem :\ Most paraboloid involving problems start by changing to polar coordinates- should I do it for this one? I know that at the end it will end up being a double integral, but I am not sure how to set it up.

physics forums have been a big help. Thanks!
 
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  • #2
Are your equations correct because z=4-y2 isn't the equation of a cylinder?
 
  • #3
vela said:
Are your equations correct because z=4-y2 isn't the equation of a cylinder?
Yes, it is. z= 4- y2 is a parabola in the yz-plane and, extended infinitely in the x-direction, is a parabolic cylinder, though not, of course, a circular cylinder.
 
  • #4
D'oh!
 
  • #5
thanks for the reply- yes, just like what hallsofivy mentioned, the equations are correct.
At this point, I am still lost, however. Any other suggestions? Thanks!
 
  • #6
As you mentioned in your original post, you want to calculate something like
[tex]V = \iint\limits_A [z_1(x,y)-z_2(x,y)]\,dy\,dx[/tex]
 
  • #7
Last edited by a moderator:
  • #8
Why are the lower limits 0 for both x and y?
 
  • #9
vela said:
Why are the lower limits 0 for both x and y?

Well I assumed so, since the problem shape starts from the 0 position for all 3 coordinate systems. Is this approach not correct?
 
  • #10
Does the original problem statement say the solid is bounded by the x=0, y=0, and z=0 planes or something equivalent? If it does, your limits look fine. I know the picture suggests this, but you never mentioned it in the original post, nor does it appear in your scan.
 
  • #11
Hey guys, I know this is bringing up an old topic, but I wanted to inquire about something, as well as make sure I approached the final equation correctly.

To clear up the confusion from vela- yes, I am planning to go with the description saying that since they said to find the volume as indicated in the picture, my limits I set up was going to be from 0.

Therefore, I went ahead and said

∫∫(2x^2-y^2)-(4-y^2),y,0,√2-x^2),x,0,√2).
After evaluating this, I obtained -pi as my answer. the number makes sense, but the sign is wrong- negative volume is obviously impossible. When I reversed the two functions, I indeed get pi as the answer.

However, that doesn't make sense- wouldn't the z1 function have to be the function of the paraboloid, and the volume is a subtraction of the cylinder function z2 from z1?

Any clarification and a check to the final answer will be appreciated. Thanks guys!
 
  • #12
iqjump123 said:
However, that doesn't make sense- wouldn't the z1 function have to be the function of the paraboloid, and the volume is a subtraction of the cylinder function z2 from z1?
Why do you think that would be the case?
 

1. How do you find the volume bounded by a paraboloid and a cylinder?

To find the volume bounded by a paraboloid and a cylinder, you can use the method of integration. First, set up the integral by determining the limits of integration and the function of the paraboloid and cylinder. Then, integrate the function with respect to the appropriate variables and evaluate the integral to find the volume.

2. What is the formula for finding the volume bounded by a paraboloid and a cylinder?

The formula for finding the volume bounded by a paraboloid and a cylinder is V = ∫A(x,y) dA, where V is the volume, A(x,y) is the function of the paraboloid and cylinder, and dA is the differential element of area.

3. Can you use different methods to find the volume bounded by a paraboloid and a cylinder?

Yes, there are multiple methods that can be used to find the volume bounded by a paraboloid and a cylinder. Some common methods include using integrals, the disk method, and the shell method. The method used may depend on the shape of the paraboloid and cylinder and personal preference.

4. Are there any special cases to consider when finding the volume bounded by a paraboloid and a cylinder?

Yes, there are a few special cases to consider when finding the volume bounded by a paraboloid and a cylinder. If the paraboloid and cylinder intersect, you will need to split the integral into separate parts. Additionally, if the paraboloid and cylinder are not symmetrical, you may need to adjust the limits of integration and use multiple integrals.

5. What real-life applications involve finding the volume bounded by a paraboloid and a cylinder?

Finding the volume bounded by a paraboloid and a cylinder has many real-life applications, including in engineering, architecture, and physics. For example, architects may use this concept when designing curved structures, and engineers may use it when calculating the volume of a curved tank or container. In physics, this concept can be used to calculate the volume of a curved object, such as a satellite dish.

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