Double Integration to Find Volume

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    Integration Volume
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Homework Help Overview

The problem involves finding the volume of a region defined by the surface z=x^2+y^2, bounded between z=0 and z=10. The context is within the subject area of calculus, specifically focusing on double integration techniques.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the necessity of using double integration for the volume calculation, with some exploring the possibility of alternative methods such as triple integration. There is also consideration of the geometric interpretation of the solid and the area of cross-sectional disks at various heights.

Discussion Status

The discussion is active, with participants offering different perspectives on how to approach the problem. Some guidance has been provided regarding the geometric interpretation of the volume, particularly in relation to the disks formed at each height.

Contextual Notes

There is a specific emphasis on using double integration, as indicated by one participant's request to avoid triple integration methods. The problem setup and the nature of the surfaces involved are also under consideration.

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Homework Statement



Find the volume of the region inside the surface z=x^2+y^2 and between z=0 and z=10

Homework Equations



x^2+y^2=10

The Attempt at a Solution



I know that I have to use some sort of double integration to find this volume, but I'm not really sure where to begin with the problem
 
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[tex]\iiint_V dV[/tex] the upper curve is [tex]z=10[/tex] the lower curve is [tex]z=0[/tex] the area of integration in [tex]\mathbb{R}^2[/tex] is a circle of radius 10. Now use cyclindrical change of variable.
 
hmm...I may try that if I can't find an alternative, but is there any way to do this problem with a double integration and not a triple integration? because the section I'm working with is strictly double integration
 
That's a paraboloid. Since you have "flat" bottom and top I recommend you imagine the solid consisting of thin horizontal pieces. It should be obvious that the piece at height z is a disk satisfying [itex]x^2+ y^2= z[/itex]. What is the area of that disk? If you think of the disk as having thickness "dz", what is its volume? Now "add" the volumes of all those disks.
 

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