Triple Integral: Volume of a Solid

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Homework Help Overview

The problem involves using a triple integral to find the volume of a solid defined by a cylinder and two planes. The cylinder is described by the equation x² + y² = 9, while the planes are given by y + z = 16 and z = 1.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to set up the bounds for the triple integral but expresses uncertainty about the limits for x. Some participants discuss the implications of the planes not intersecting within the cylinder and suggest that this simplifies the parametrization of the integral.

Discussion Status

Participants are exploring the setup of the integral and discussing the implications of the geometric configuration of the solid. There is a recognition of the challenges in determining the bounds, particularly for the x variable. Some guidance has been offered regarding the relationship between the planes and the cylinder.

Contextual Notes

The original poster notes difficulty in setting up the bounds of the integral, particularly for the x dimension, and questions whether their proposed limits are correct. There is also a mention of the planes not intersecting within the cylinder, which may affect the setup.

iamalexalright
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Homework Statement


Been awhile since I've done them and my memory/reasoning isn't so great apparently...

Use the triple integral to find the volume of the given solid:
The solid enclosed by the cylinder
[tex]x^{2} + y^{2} = 9[/tex]
and the planes y + z = 16 and z = 1. 2. The attempt at a solution
Difficulty is always setting up the bounds of the integral...
[tex]-3 \leq y \leq 3[/tex]
[tex]1 \leq z \leq 16 - y[/tex]
having problems with the xwould it be:
[tex]-\sqrt{9 - y^{2}} \leq x \leq \sqrt{9 - y^{2}}[/tex] ?
 
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Sure. The planes don't intersect inside the cylinder. So you can parametrize the integral over the x,y in the circle defining the cylinder without worrying about the z value. If the planes had intersected inside the circle they would have had to give you a more elaborate description of the region.
 
135*pi, cool! Thanks Dick
 
iamalexalright said:
135*pi, cool! Thanks Dick

That's what I get. :)
 

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