Sketching and Calculating the Volume of Solid E

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Homework Help Overview

The discussion revolves around sketching and calculating the volume of a solid E defined by the cylinder x = y^2 and the planes z = 3 and x + z = 1. Participants are attempting to visualize the solid and express it analytically while exploring the implications of the given boundaries.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants are sharing their attempts at sketching the solid and questioning the feasibility of enclosing a volume with the given boundaries. There are discussions about the intersections of the planes and the cylinder, as well as the implications of the equations involved.

Discussion Status

The discussion is ongoing, with participants providing insights and questioning the assumptions made in the problem setup. Some have suggested that an upper limit on x may be necessary for the solid to be properly defined, indicating a productive exploration of the problem's constraints.

Contextual Notes

There are indications that some participants feel there may be missing information that could clarify the boundaries of the solid, particularly regarding the upper limit on x.

kieranl
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Homework Statement


Sketch the solid E bounded by the cylinder x = y^2 and the planes z = 3 and x + z = 1, and write down its analytic expression. Then, use a triple integral to find the volume of E.

The Attempt at a Solution


Was wondering if someone could have a go at drawing this sketch? In mine, I thought i did it right but can't seem to obtain an enclosed surface. If x+z=1 was rather x-z=1 i would be able to but can't so far?
 
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The plane x+ z= 1 crosses the plane z= 3 when x+ 3= 1 or the line x= -2, z= 3, y= t. It crosses the cylinder x= y^2 in the line x= t^2, y= t, z= 1- x= 1- t^2.

I wonder if you weren't confusing x= y^2 with the y= x^3.
 
I don't think you NEED to graph this.

y=(plus/minus) sqrt(x)

If that helps.

The range is x>0, so sqrt(x) is real.

If x>0, which is on top, z=3 or z=1-x?EDIt: Is there an upper bound on x?
 
wat i have done is drawn the cylinder x = y^2 in the x-y plane and extended it along the z plane. Then i drew the z=3 plane and then drew a line z=1-x and it extended it along the y plane. But this does not end up enclosing a solid
 
kieranl said:
wat i have done is drawn the cylinder x = y^2 in the x-y plane and extended it along the z plane. Then i drew the z=3 plane and then drew a line z=1-x and it extended it along the y plane. But this does not end up enclosing a solid

You are right, I think. There must be an upper x-limit for this to be a solid. Maybe, you have accidentally skipped some information.
 

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