Volumes of Revolution: Revolving about a vertical axis

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SUMMARY

The discussion focuses on calculating the volume of the solid formed by rotating the area between the curves y=1/x and y=1/x^2 around the vertical axis x=-1. The volume of revolution formula, V = π ∫_a^b (R^2 - r^2) dx, is utilized, where R and r represent the outer and inner radii, respectively. The outer radius R is determined as 1 - (1/x^2) and the inner radius r as 1 - (1/x). The integration limits are established as a=2 and b=5 based on the y-axis points (0,2) and (0,5).

PREREQUISITES
  • Understanding of volume of revolution concepts
  • Familiarity with the functions y=1/x and y=1/x^2
  • Knowledge of integration techniques
  • Ability to visualize solid rotation about an axis
NEXT STEPS
  • Study the application of the volume of revolution formula in different scenarios
  • Learn about the method of cylindrical shells for volume calculations
  • Explore the concept of definite integrals in calculus
  • Investigate graphical representations of functions and their rotations
USEFUL FOR

Students studying calculus, particularly those focusing on volume calculations and solid geometry, as well as educators teaching these concepts in mathematics courses.

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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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.
 

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