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

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SUMMARY

The discussion focuses on methods for calculating the volume of solids of revolution, specifically addressing three problems: finding the volume of a solid generated by revolving the circle defined by the equation x² + (y - 1)² = 1 about the y-axis, using the cylindrical method for a sphere of radius r, and determining the volume of a right circular cone with radius r and height h. The volume of the sphere is established as V = (4/3)πr³, and the disk method is suggested for calculating the volume of the solid generated by the first problem.

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  • Understanding of solid geometry and volume calculations
  • Familiarity with the disk and shell methods for finding volumes of solids of revolution
  • Knowledge of the equation of a circle in Cartesian coordinates
  • Basic calculus concepts, including integration
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  • Study the disk method for calculating volumes of solids of revolution
  • Learn the cylindrical method for finding the volume of a sphere
  • Explore the formula for the volume of a right circular cone: V = (1/3)πr²h
  • Investigate applications of the washer method for more complex solids of revolution
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Students and educators in mathematics, particularly those studying calculus and solid geometry, as well as professionals involved in engineering and design requiring volume calculations of three-dimensional shapes.

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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