Simpsons rule volume washers shells about y=-1

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Discussion Overview

The discussion revolves around using Simpson's rule to approximate the volume of a solid obtained by rotating a region bounded by the curves y=x^3, y=1, and x=0 about the line y=-1. Participants explore the integration methods and the application of Simpson's rule in this context, particularly focusing on the challenges posed by the rotation axis not being a principal axis.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • One participant asks for the volume integral to use and how to express it in Simpson's rule form.
  • Another participant suggests sketching the region and shifting the origin as potential strategies to approach the problem.
  • A participant expresses familiarity with solids of rotation and Simpson's rule but struggles to combine them due to the rotation about y=-1, questioning how to handle the integration when the integral is in y's.
  • There is a suggestion that the position and orientation of the axes do not matter, and that moving to a new coordinate system aligned with the rotation axis could simplify the problem.

Areas of Agreement / Disagreement

Participants do not reach a consensus, as there are differing views on how to approach the integration and the application of Simpson's rule in this specific scenario. The discussion remains unresolved regarding the best method to combine these techniques.

Contextual Notes

Participants note the potential complexity introduced by the rotation axis being y=-1, which may affect the integration process. There is also mention of the need to shift the coordinate system, but the implications of this shift on the integration variables remain unclear.

z37002
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Simpsons rule n=6, to approximate volume of solid obtained by rotating the region bounded by y=x^3,y=1,x=0 about line y=-1. What is the volume integral to use, and how is it put into simpsons rule form? thanks.
 
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Welcome to PF;
Those are questions for you ... how have you attempted to answer them so far?
Hint:
State Simpson's rule
Sketch the region
Shift the origin

Have you studied solids of rotation before?
 
Yes, I've done both revolution of solid and simpsons, but not together? And it is the about y=-1 as opposed to on the axis that I seek help w. And if the integral I know is shells, than how do plug in x values if integral is in y's? I am not seeing this as being able to be simple discs. Though a shift of y-axis shouldn't effect x values, I still see the y=-1 as a deterrent to combining simpsons sand shells. IF another way or similar example please share.
 
Any advice would be good.
 
z37002 said:
Any advice would be good.
Post a link to your sketch.
 
You know how to do solids of rotation about one of the principle axes of the coordinate system.
However, you don't know how to handle arbitrary rotations?

What you have to do is realize that the positon and orientation of the axes doesn't matter for the problem - so move to a new coordinate system where one of the principle axes lines up with the rotation axis.

You should see how to do this if you sketch the situation out.
 

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