Volumes of Revolution: Revolving about a vertical axis

In summary: The volume is the integral from y= 2 to y= 5 of pi[(1- 1/x2)2- (1- 1/x)2]dy.In summary, the problem involves finding the volume of a solid obtained by rotating the region between the graphs of y=1/x and y=1/x^2 around the line x=1. The outer and inner radii are given as R=1-1/x^2 and r=1- 1/x, respectively, and the volume can be calculated using the integral from y=2 to y=5 of pi[(1- 1/x^2)^2- (1- 1/x)^2]dy.
  • #1
Beeorz
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Homework Statement


Find the volume of the solid obtained by rotating the region under the graph of y=1/x and y=1/x^2 about the vertical axis x=-1

Also given are points on the y-axis (0,2) and (0,5). I guessing these points are specific sections of the graphs where we find the volume.



Homework Equations


Volume of revolution = pi int_a^b ((R^2)-(r^2))dx


The Attempt at a Solution


For this problem I would think...
a=2
b=5
but I am uncertain as to how to find R and r. I know they correspond the outer and inner radii but how do I go about find them.

I attached a picture of what the problem looks like.
 

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  • #2
Beeorz said:

Homework Statement


Find the volume of the solid obtained by rotating the region under the graph of y=1/x and y=1/x^2 about the vertical axis x=-1
Not "under" the graphs, between them.

Since you drew a picture of the curves, also draw the line x= 1 and imagine a horizontal line from from x= 1 to those curves, for a given y.

Also given are points on the y-axis (0,2) and (0,5). I guessing these points are specific sections of the graphs where we find the volume.



Homework Equations


Volume of revolution = pi int_a^b ((R^2)-(r^2))dx


The Attempt at a Solution


For this problem I would think...
a=2
b=5
but I am uncertain as to how to find R and r. I know they correspond the outer and inner radii but how do I go about find them.

I attached a picture of what the problem looks like.
You are rotating the outer graph, y= 1/x2, around the line x= 1. R is the distance between those two, for each y: 1- 1/x2. r, the inner radius, is the distance from the curve y= 1/x to 1: 1- 1/x.
 

Related to Volumes of Revolution: Revolving about a vertical axis

1. What is a volume of revolution?

A volume of revolution is the three-dimensional space created when a two-dimensional shape is rotated around a fixed axis. This shape is called the "generating curve" and the axis is known as the "axis of revolution."

2. What does it mean to revolve about a vertical axis?

Revolving about a vertical axis means that the generating curve is rotated around a vertical line, such as the y-axis on a graph. This results in a three-dimensional shape that is symmetrical around the vertical axis.

3. How is the volume of revolution calculated?

The volume of revolution is calculated using calculus and the method of cylindrical shells or the method of disks/washers. These methods involve integrating the cross-sectional area of the shape as it is rotated around the axis of revolution.

4. What types of shapes can be used to create a volume of revolution?

Any two-dimensional shape can be used to create a volume of revolution, as long as it is rotated around a fixed axis. Common shapes used include circles, rectangles, triangles, and more complex curves such as parabolas or ellipses.

5. What are some real-life applications of volumes of revolution?

Volumes of revolution have many practical applications in fields such as engineering, architecture, and physics. For example, they can be used to calculate the volume of a bottle or a tire, the amount of material needed to create a specific shape, or the amount of fluid in a container with a curved surface.

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