What are the methods for finding the volume of a solid of revolution?

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Discussion Overview

The discussion revolves around methods for finding the volume of solids of revolution, specifically through three problems involving a circle, a sphere, and a cone. The focus includes theoretical approaches and mathematical reasoning related to these shapes.

Discussion Character

  • Homework-related, Mathematical reasoning

Main Points Raised

  • One participant presents three problems related to finding volumes of solids of revolution, including a circle about the y-axis, a sphere, and a cone.
  • Another participant identifies the region described by the equation as a circle centered at (0, 1) with a radius of 1 unit.
  • It is noted that revolving the circle around an axis through its center will result in a sphere with the same radius, leading to an expected volume of $$V=\frac{4}{3}\pi$$.
  • Participants discuss using either the disk or shell method for calculating the volume, with a suggestion to start with the disk method.

Areas of Agreement / Disagreement

Participants agree on the identification of the circle and its properties, but the discussion on methods and calculations remains open and unresolved.

Contextual Notes

There are no explicit limitations noted, but the discussion may depend on the participants' understanding of the methods for calculating volumes.

Who May Find This Useful

Students or individuals interested in calculus, specifically in the study of solids of revolution and volume calculation methods.

annie122
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i have no clue as to how to proceed for the following three problems:

#1 find the volume of the resulting solid by any method
x^2 + (y - 1)^2 = 1, about the y -axis

#2 use the cylindrical method to obtain the volume of
a sphere of radius r

#3 and a right circular cone of radius r and height h

thank you
 
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We ask that you post no more than 2 questions per thread, so just keep this in mind for the future. These are all related questions, so we can leave all three here.

Let's begin with the first question. What region is described by the given equation?
 
sorry :(

a circle with center (0, 1).
 
Yuuki said:
sorry :(

a circle with center (0, 1).

Yes, with a radius of 1 unit. Since the center of this circle is on the $y$-axis, revolving the circle around an axis passing through the center will result in a sphere having the same radius as the circle. So, we should expect the volume of this solid of revolution to have a measure of $$V=\frac{4}{3}\pi$$.

We can use either the disk or shell method. Let's try the disk method first. Can you state the volume of an arbitrary disk?
 

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