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

[annie122]

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

 

[MarkFL]

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?

 

[annie122]

sorry :(

a circle with center (0, 1).

 

[MarkFL]

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?