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

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In summary, the given equation describes a circle with center (0, 1) and a radius of 1 unit. Revolving this circle around the y-axis will result in a sphere with a volume of V=\frac{4}{3}\pi. The volume can be calculated using the disk or shell method.
  • #1
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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  • #2
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?
 
  • #3
sorry :(

a circle with center (0, 1).
 
  • #4
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 \(\displaystyle V=\frac{4}{3}\pi\).

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?
 

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

What is volume and how is it measured?

Volume is the amount of space occupied by an object or substance. It is typically measured in units such as cubic meters (m3) for solid objects, liters (L) for liquids, and cubic centimeters (cm3) for gases.

How does temperature affect volume?

Temperature can affect volume in different ways depending on the substance. For most substances, an increase in temperature causes the particles to move faster and take up more space, resulting in an increase in volume. However, for water, there is a unique temperature at which it is most dense, causing it to have a slightly lower volume compared to other temperatures.

How does pressure affect volume?

Pressure is the force applied to a unit of area. In general, an increase in pressure causes a decrease in volume, as the particles are forced closer together. This is known as Boyle's Law. However, for gases, an increase in pressure can also cause an increase in volume if the temperature is also increased, as stated by Charles's Law.

What is the relationship between volume and mass?

Volume and mass are two different measurements, but they are related in some cases. For example, the density of a substance is the mass per unit volume. This means that as the volume of a substance increases, its mass will also increase if the density remains constant.

How can I calculate the volume of an irregularly shaped object?

To calculate the volume of an irregularly shaped object, you can use the water displacement method. Place the object in a container of water and measure the volume of water before and after the object is submerged. The difference between the two volumes is the volume of the object. Alternatively, you can also use calculus to find the volume of complex shapes by integrating cross-sectional areas along its length.

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