Finding Volume of Solid Generated by Revolving Cycloid Arch Around x-Axis

In summary: Using substitution, we can rewrite dV as (pi)(1-cos(theta))^2 [1-cos(theta)]dtheta. From there, we can integrate to find the volume of the solid generated by revolving the region bounded by the x-axis and one arch of the cycloid, resulting in a much simpler integral. In summary, the volume of the solid can be found using the hint given and by using substitution to simplify the integral.
  • #1
Yuma
3
0
1. Find the volume of the solid generated by revolving the region bounded by the x-axis and one arch of the cycloid x=theta-sin(theta), y=1-cos(theta) around the x-axis.



2. hint:dV=(pi)y^2 dx



3. So far I have been unable to solve for theta so that I can form a relationship between the two equations. I don't believe that solving the x= equation for theta can be done, and the integral that I come up with if I solve the y= equation for theta and substitute it into the x= equation is so unwieldy that I don't believe it is the right one. Besides, the shape is revolving around the x-axis, so I should have a y= equation in order to proceed with solving this problem in the way that I was taught. Any help is much appreciated!
 
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  • #2
Yuma said:
1. Find the volume of the solid generated by revolving the region bounded by the x-axis and one arch of the cycloid x=theta-sin(theta), y=1-cos(theta) around the x-axis.



2. hint:dV=(pi)y^2 dx



3. So far I have been unable to solve for theta so that I can form a relationship between the two equations. I don't believe that solving the x= equation for theta can be done, and the integral that I come up with if I solve the y= equation for theta and substitute it into the x= equation is so unwieldy that I don't believe it is the right one. Besides, the shape is revolving around the x-axis, so I should have a y= equation in order to proceed with solving this problem in the way that I was taught. Any help is much appreciated!


dV= (pi)y^2 dx. y= 1- cos(theta) and, since x= theta- sin(theta), dx= [1- cos(theta)]dtheta.
 

1. What is a "Solids of Revolution Problem"?

A "Solids of Revolution Problem" is a type of mathematical problem in which a three-dimensional object is created by rotating a two-dimensional shape around an axis. This creates a solid shape, such as a cone, cylinder, or sphere.

2. How is the volume of a solid of revolution calculated?

The volume of a solid of revolution can be calculated using the formula V = π∫(f(x))^2 dx, where f(x) is the function that defines the shape being rotated and the integral is taken over the desired interval.

3. What are some real-world applications of solids of revolution?

Solids of revolution have many practical applications, such as in engineering, architecture, and design. For example, they can be used to model objects like bottles, pipes, and propellers.

4. What are some common methods for solving solids of revolution problems?

Some common methods for solving solids of revolution problems include using the disk or washer method, the shell method, and the cylindrical method. These methods involve breaking the shape down into smaller, simpler shapes and using integration to calculate the volume.

5. Are there any important considerations when solving solids of revolution problems?

Yes, there are several important considerations to keep in mind when solving solids of revolution problems. These include correctly identifying the axis of rotation, choosing the appropriate integration method, and carefully setting up the integral with the correct bounds and function to be integrated.

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