Find the volume of the region bounded by parabolic cylinder and planes

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Homework Help Overview

The problem involves finding the volume of a solid bounded by a parabolic cylinder defined by the equation y = x^2 and the planes z = 3 - y and z = 0. Participants are discussing the setup for a triple integral to determine the limits of integration for the variables involved.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants are attempting to establish the limits of integration for z, y, and x. There is confusion regarding how to derive the limits for x and y based on the given equations. Some suggest visualizing the solid and sketching it to better understand the region of integration.

Discussion Status

The discussion is ongoing, with participants offering guidance on visualizing the solid and questioning the accuracy of the proposed limits. There is an emphasis on the need to graph the region to clarify the integration boundaries.

Contextual Notes

Participants note that the professor did not explain how to find the limits of integration, which is contributing to the confusion. There is also a mention of needing to consider the intersection of the planes and the x-y plane.

mharten1
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Homework Statement



Find the volume of the solid bounded by the parabolic cylinder y = x^2 and the planes z = 3-y and z = 0

Homework Equations


The Attempt at a Solution


Obviously, a triple integral must be used in the situation. Our professor never explained how to find the limits of integration, this is the part that is confusing me. I believe that the limits for z are 3-y and 0. However, I have no idea how to get the limits of x and y. Any help is greatly appreciated.
 
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hi mharten1! :smile:
mharten1 said:
… the limits for z are 3-y and 0. However, I have no idea how to get the limits of x and y. Any help is greatly appreciated.

no, your first limits (for z, say) will just be two numbers (no y)

your next limits (for y, say) will depend on z

your third limits will depend on y and z

standard method: cut the region into horizontal slices between heights z and z+dz …

what is the 2D shape of each slice? :wink:​
 
Basically you want to draw a sketch of the solid you are trying to find the volume of. Can you visualize it? z=0 is the x-y plane. Where does the plane z=y-3 intersect the x-y plane?
 
Dick said:
Basically you want to draw a sketch of the solid you are trying to find the volume of. Can you visualize it? z=0 is the x-y plane. Where does the plane z=y-3 intersect the x-y plane?

Doesn't it intersect it when z=0? So at y=3?
 
mharten1 said:
Doesn't it intersect it when z=0? So at y=3?

Sure. So what region do you want to integrate z over in the x-y plane?
 
Dick said:
Sure. So what region do you want to integrate z over in the x-y plane?

The region from 0 to 3? If that's not right, I guess I'm not visualizing this in the right way. I think I'll graph out the region so I can better see it.
 
mharten1 said:
The region from 0 to 3? If that's not right, I guess I'm not visualizing this in the right way. I think I'll graph out the region so I can better see it.

I think you should graph it out. You should anyway. "The region from 0 to 3?" isn't a very accurate description.
 

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