Disk, Washer, Shell Multiple Integrals

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Homework Help Overview

The discussion revolves around determining the number of integrals required for the disk, washer, and shell methods in the context of calculating volumes of solids of revolution. The specific functions involved are x=3y^2 - 2 and x=y^2, with the region of interest defined from (-2,0) to (1,1) about the x-axis.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • Participants discuss the necessity of multiple integrals for the disk and washer methods, with some suggesting that the lack of breaks in the graph might imply only one integral is needed. Questions are raised about how to determine when more than one integral is required, and the axis of revolution is clarified.

Discussion Status

The conversation is ongoing, with participants exploring different interpretations of the problem. Some guidance has been offered regarding the need for sketches to visualize the region and the resulting solid, which may clarify the necessity for multiple integrals in certain cases.

Contextual Notes

There is a mention of the need to consider changes in shape at specific points when using disks or washers, indicating that the problem may involve complexities that require careful analysis of the region being revolved.

whatlifeforme
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Homework Statement


Determine how many integrals are required for disk, washer, and shell method.

Homework Equations


x=3y^2 - 2 and x=y^2 from (-2,0) to (1,1) about x-axis.

The Attempt at a Solution


Since there are no breaks or abnormalities in the graph it appears that 1 integral will solve for each to me. I have no other way of looking at this problem.

how do i know when the disk method requires more than one integral?
 
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whatlifeforme said:

Homework Statement


Determine how many integrals are required for disk, washer, and shell method.

Homework Equations


x=3y^2 - 2 and x=y^2 from (-2,0) to (1,1)

The Attempt at a Solution


Since there are no breaks or abnormalities in the graph it appears that 1 integral will solve for each to me. I have no other way of looking at this problem.

how do i know when the disk method requires more than one integral?
What axis is this to be revolved around ?
 
SammyS said:
What axis is this to be revolved around ?

sorry. about the x-axis.

evidently it will take two for disk and washer, and one for shell, but I'm not sure why.
 
whatlifeforme said:
sorry. about the x-axis.

evidently it will take two for disk and washer, and one for shell, but I'm not sure why.
Have you drawn a sketch of the region that will be rotated?
Have you drawn a different sketch of the solid that results from the rotation?

If you haven't drawn both sketches, it will be very difficult to work this problem.

If you use disks/washers of width Δx, your 2nd drawing should show that they change shape at (0, 0), because the upper and lower edges change. This means you will need two integrals.

If you use cylindrical shells of width Δy, the left and right edges don't change, so a single integral will suffice.
 

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