Volume of Solid in First Octant: Triple Integration Problem

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Homework Help Overview

The problem involves finding the volume of a solid in the first octant, defined by the surfaces z=1-y², y=2x, and the plane x=3. The original poster expresses uncertainty about visualizing the region for integration and whether graphing is necessary for determining limits.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the need for graphing to understand the integration limits and the relationships between the variables. There is an attempt to write out the integral based on the given constraints, but questions arise regarding the correct order of integration and the dependencies of the bounds.

Discussion Status

Some participants have provided guidance on the order of integration and the necessity of incorporating the constraints as bounds in the integrals. There is an ongoing exploration of how to properly set up the integral based on the relationships between the variables.

Contextual Notes

Participants note that the bounds for y depend on x, and the bounds for z depend on y, indicating a need to carefully consider these dependencies in the integration setup.

aaronfue
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Homework Statement



Find the volume of the solid in the first octant bounded by the graphs of:
z=1-y2
y=2x
x=3

Homework Equations



I was able to graph all three but I can't picture the region for integration. I'm not sure if I even have to graph it or if I can get my limits without the graph.

The Attempt at a Solution

 
Last edited:
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Please post an attempt at writing out the integral.
 
After graphing the equations, I came up with:

\int^{1}_{-1} \int^{3}_{0} \int^{1}_{0} dzdxdy (this order was given as a hint)

My final answer was 6?
 
No, you need to put the given constraints as bounds in the integrals. The order needed also results from the dependencies in those bounds. The y bounds depend on x, and the z bounds depend on y, and therefore on x too. So the order should be z, y, x.
 

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