Volume of Solid Between Curves: Region, Sketch, and Solution

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Homework Help Overview

The discussion revolves around finding the volume of a solid obtained by rotating a region bounded by specific curves about a designated line. The subject area includes calculus concepts related to volume of solids of revolution, specifically using disk and washer methods.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the need for clarity regarding the line of rotation and the specific curves involved. There is uncertainty about the correct representation of the solid and typical disks or washers. Questions arise about the completeness of the problem statement and the interpretation of the equations provided.

Discussion Status

The discussion is ongoing, with some participants suggesting that the original poster clarify the line of rotation and the specific details of the problems. There is a recognition that additional information is necessary for a complete understanding, and some guidance has been offered regarding the orientation of disks and washers.

Contextual Notes

Participants note that the original poster's problems lack critical information, such as the line around which the region is rotated. This missing context may hinder the ability to provide effective assistance.

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Homework Statement



Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

Homework Equations





The Attempt at a Solution



Sometimes, I struggled with drawing a solid and a typical disc. Also, I am unsure with the solution for problem 4. Do you see any mistakes?

I would so much appreciate your help.
 

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This is a rare time when no one replied. Sad.
 
You'll probably do better submitting one problem at a time.

The first couple of problems are missing some information - namely, the line around which the region is rotated. For example, in #2, you have the region bounded by y = 1 - x2 and y = 0, but you didn't say what this region is rotated around.
 
It's probably rotated around y = 0, especially since the problem mentions "disk and washer."

I'm guessing the cylinder shape you drew was supposed to be a typical disk/washer the problem asked for, but that's not what you want. When the region is rotated around a horizontal line, disks and washers will be vertical, between the axis of revolution and the function for disks or between the two given functions for washers.

On #4, what are the last three equations for?
y = 0, x = 2, x = 4
 
Bohrok said:
It's probably rotated around y = 0, especially since the problem mentions "disk and washer."
Disks and washers can be oriented either vertically or horizontally.
 

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